Basic formulas of electrodynamics. Physics formulas for the exam

The session is approaching, and it's time for us to move from theory to practice. Over the weekend, we sat down and thought that many students would do well to have a collection of basic physics formulas handy. Dry formulas with explanation: short, concise, nothing more. A very useful thing when solving problems, you know. Yes, and at the exam, when exactly what was cruelly memorized the day before can “jump out” of my head, such a selection will serve you well.

Most of the tasks are usually given in the three most popular sections of physics. This is Mechanics, thermodynamics and Molecular physics, electricity. Let's take them!

Basic formulas in physics dynamics, kinematics, statics

Let's start with the simplest. Good old favorite rectilinear and uniform movement.

Kinematic formulas:

Of course, let's not forget about the movement in a circle, and then move on to the dynamics and Newton's laws.

After the dynamics, it's time to consider the conditions for the equilibrium of bodies and liquids, i.e. statics and hydrostatics

Now we give the basic formulas on the topic "Work and Energy". Where would we be without them!

Basic formulas of molecular physics and thermodynamics

Let's finish the section of mechanics with formulas for vibrations and waves and move on to molecular physics and thermodynamics.

Efficiency, Gay-Lussac's law, the Clapeyron-Mendeleev equation - all these sweet formulas are collected below.

By the way! There is a discount for all our readers 10% on the any kind of work.

Basic formulas in physics: electricity

It's time to move on to electricity, although thermodynamics loves it less. Let's start with electrostatics.

And, to the drum roll, we finish with the formulas for Ohm's law, electromagnetic induction and electromagnetic oscillations.

That's all. Of course, a whole mountain of formulas could be given, but this is useless. When there are too many formulas, you can easily get confused, and then completely melt the brain. We hope that our cheat sheet of basic formulas in physics will help you solve your favorite problems faster and more efficiently. And if you want to clarify something or have not found the formula you need: ask the experts student service. Our authors keep hundreds of formulas in their heads and click tasks like nuts. Contact us, and soon any task will be "too tough" for you.

Cheat sheet with formulas in physics for the exam

And not only (may need 7, 8, 9, 10 and 11 classes). For starters, a picture that can be printed in a compact form.

A cheat sheet with formulas in physics for the Unified State Examination and not only (grades 7, 8, 9, 10 and 11 may need it).

and not only (may need 7, 8, 9, 10 and 11 classes).

And then the Word file, which contains all the formulas to print them, which are at the bottom of the article.

Mechanics

Pressure P=F/S
Density ρ=m/V
Pressure at the depth of the liquid P=ρ∙g∙h
Gravity Ft=mg
5. Archimedean force Fa=ρ w ∙g∙Vt
Equation of motion for uniformly accelerated motion

X=X0 + υ 0∙t+(a∙t 2)/2 S=( υ 2 -υ 0 2) /2а S=( υ +υ 0) ∙t /2

Velocity equation for uniformly accelerated motion υ =υ 0 +a∙t
Acceleration a=( υ -υ 0)/t
Circular speed υ =2πR/T
Centripetal acceleration a= υ 2/R
Relationship between period and frequency ν=1/T=ω/2π
Newton's II law F=ma
Hooke's law Fy=-kx
Law of universal gravitation F=G∙M∙m/R 2
The weight of a body moving with acceleration a P \u003d m (g + a)
The weight of a body moving with acceleration a ↓ P \u003d m (g-a)
Friction force Ffr=µN
Body momentum p=m υ
Force impulse Ft=∆p
Moment M=F∙ℓ
Potential energy of a body raised above the ground Ep=mgh
Potential energy of elastically deformed body Ep=kx 2 /2
Kinetic energy of the body Ek=m υ 2 /2
Work A=F∙S∙cosα
Power N=A/t=F∙ υ
Efficiency η=Ap/Az
Oscillation period of the mathematical pendulum T=2π√ℓ/g
Oscillation period of a spring pendulum T=2 π √m/k
The equation of harmonic oscillations Х=Хmax∙cos ωt
Relationship of the wavelength, its speed and period λ= υ T

Molecular physics and thermodynamics

Amount of substance ν=N/ Na
Molar mass M=m/ν
Wed. kin. energy of monatomic gas molecules Ek=3/2∙kT
Basic equation of MKT P=nkT=1/3nm 0 υ 2
Gay-Lussac law (isobaric process) V/T =const
Charles' law (isochoric process) P/T =const
Relative humidity φ=P/P 0 ∙100%
Int. ideal energy. monatomic gas U=3/2∙M/µ∙RT
Gas work A=P∙ΔV
Boyle's law - Mariotte (isothermal process) PV=const
The amount of heat during heating Q \u003d Cm (T 2 -T 1)
The amount of heat during melting Q=λm
The amount of heat during vaporization Q=Lm
The amount of heat during fuel combustion Q=qm
The equation of state for an ideal gas is PV=m/M∙RT
First law of thermodynamics ΔU=A+Q
Efficiency of heat engines η= (Q 1 - Q 2) / Q 1
Ideal efficiency. engines (Carnot cycle) η \u003d (T 1 - T 2) / T 1

Electrostatics and electrodynamics - formulas in physics

Coulomb's law F=k∙q 1 ∙q 2 /R 2
Electric field strength E=F/q
Email tension. field of a point charge E=k∙q/R 2
Surface charge density σ = q/S
Email tension. fields of the infinite plane E=2πkσ
Dielectric constant ε=E 0 /E
Potential energy of interaction. charges W= k∙q 1 q 2 /R
Potential φ=W/q
Point charge potential φ=k∙q/R
Voltage U=A/q
For a uniform electric field U=E∙d
Electric capacity C=q/U
Capacitance of a flat capacitor C=S∙ ε ∙ε 0/d
Energy of a charged capacitor W=qU/2=q²/2С=CU²/2
Current I=q/t
Conductor resistance R=ρ∙ℓ/S
Ohm's law for the circuit section I=U/R
The laws of the last compounds I 1 \u003d I 2 \u003d I, U 1 + U 2 \u003d U, R 1 + R 2 \u003d R
Parallel laws. conn. U 1 \u003d U 2 \u003d U, I 1 + I 2 \u003d I, 1 / R 1 + 1 / R 2 \u003d 1 / R
Electric current power P=I∙U
Joule-Lenz law Q=I 2 Rt
Ohm's law for a complete chain I=ε/(R+r)
Short circuit current (R=0) I=ε/r
Magnetic induction vector B=Fmax/ℓ∙I
Ampere Force Fa=IBℓsin α
Lorentz force Fл=Bqυsin α
Magnetic flux Ф=BSсos α Ф=LI
Law of electromagnetic induction Ei=ΔФ/Δt
EMF of induction in moving conductor Ei=Вℓ υ sinα
EMF of self-induction Esi=-L∙ΔI/Δt
The energy of the magnetic field of the coil Wm \u003d LI 2 / 2
Oscillation period count. contour T=2π ∙√LC
Inductive reactance X L =ωL=2πLν
Capacitance Xc=1/ωC
The current value of the current Id \u003d Imax / √2,
RMS voltage Ud=Umax/√2
Impedance Z=√(Xc-X L) 2 +R 2

Optics

The law of refraction of light n 21 \u003d n 2 / n 1 \u003d υ 1 / υ 2
Refractive index n 21 =sin α/sin γ
Thin lens formula 1/F=1/d + 1/f
Optical power of the lens D=1/F
max interference: Δd=kλ,
min interference: Δd=(2k+1)λ/2
Differential grating d∙sin φ=k λ

The quantum physics

Einstein's formula for the photoelectric effect hν=Aout+Ek, Ek=U ze
Red border of the photoelectric effect ν to = Aout/h
Photon momentum P=mc=h/ λ=E/s

Physics of the atomic nucleus

Law of radioactive decay N=N 0 ∙2 - t / T
Binding energy of atomic nuclei

E CB \u003d (Zm p + Nm n -Mya)∙c 2

HUNDRED

t \u003d t 1 / √1-υ 2 / c 2
ℓ=ℓ 0 ∙√1-υ 2 /c 2
υ 2 \u003d (υ 1 + υ) / 1 + υ 1 ∙υ / c 2
E = m with 2

Definition 1

Electrodynamics is a huge and important area of physics that studies the classical, non-quantum properties of the electromagnetic field and the motion of positively charged magnetic charges interacting with each other through this field.

Figure 1. Briefly about electrodynamics. Author24 - online exchange of student papers

Electrodynamics is represented by a wide range of various problem statements and their competent solutions, approximate methods and special cases, which are united into one whole by general initial laws and equations. The latter, constituting the bulk of classical electrodynamics, are presented in detail in Maxwell's formulas. Currently, scientists continue to study the principles of this field in physics, the skeleton of its relationship with other scientific areas.

Coulomb's law in electrodynamics is denoted as follows: $F= \frac (kq1q2) (r2)$, where $k= \frac (9 \cdot 10 (H \cdot m)) (Kl)$. The electric field strength equation is written as follows: $E= \frac (F)(q)$, and the flux of the magnetic field induction vector is $∆Ф=В∆S \cos (a)$.

In electrodynamics, first of all, free charges and systems of charges are studied, which contribute to the activation of a continuous energy spectrum. The classical description of the electromagnetic interaction is favored by the fact that it is effective already in the low-energy limit, when the energy potential of particles and photons is small compared to the rest energy of the electron.

In such situations, there is often no annihilation of charged particles, since there is only a gradual change in the state of their unstable motion as a result of the exchange of a large number of low-energy photons.

Remark 1

However, even at high energies of particles in a medium, despite the significant role of fluctuations, electrodynamics can be successfully used for a comprehensive description of average statistical, macroscopic characteristics and processes.

Basic equations of electrodynamics

The main formulas that describe the behavior of an electromagnetic field and its direct interaction with charged bodies are Maxwell's equations, which determine the probable actions of a free electromagnetic field in a medium and vacuum, as well as the general generation of a field by sources.

Among these positions in physics it is possible to distinguish:

the Gauss theorem for the electric field - designed to determine the generation of an electrostatic field by positive charges;
the hypothesis of closed field lines - promotes the interaction of processes within the magnetic field itself;
Faraday's law of induction - establishes the generation of electric and magnetic fields by variable properties of the environment.

In general, the Ampère-Maxwell theorem is a unique idea about the circulation of lines in a magnetic field with the gradual addition of displacement currents introduced by Maxwell himself, precisely determines the transformation of a magnetic field by moving charges and the alternating action of an electric field.

Charge and force in electrodynamics

In electrodynamics, the interaction of the force and charge of an electromagnetic field proceeds from the following joint definition of the electric charge $q$, energy $E$ and magnetic $B$ fields, which are approved as a fundamental physical law based on the entire set of experimental data. The formula for the Lorentz force (within the idealization of a point charge moving at a certain speed) is written with the change of speed $v$.

Conductors often contain a huge amount of charges, therefore, these charges are quite well compensated: the number of positive and negative charges is always equal to each other. Therefore, the total electrical force that constantly acts on the conductor is also equal to zero. The magnetic forces that operate on individual charges in the conductor, as a result, are not compensated, because in the presence of a current, the velocities of the charges are always different. The equation of action of a conductor with current in a magnetic field can be written as follows: $G = |v ⃗ |s \cos(a) $

If we study not a liquid, but a full-fledged and stable flow of charged particles as a current, then the entire energy potential passing linearly through the area in $1s$ will be the current strength equal to: $I = ρ| \vec (v) |s \cos(a) $, where $ρ$ is the charge density (per unit volume in the total flow).

Remark 2

If the magnetic and electric fields systematically change from point to point on a specific site, then in the expressions and formulas for partial flows, as in the case of a liquid, the average values $E ⃗ $ and $B ⃗$ on the site are necessarily put down.

Special position of electrodynamics in physics

The significant position of electrodynamics in modern science can be confirmed by the well-known work of A. Einstein, in which the principles and foundations of the special theory of relativity were detailed. The scientific work of an outstanding scientist is called "On the Electrodynamics of Moving Bodies", and includes a huge number of important equations and definitions.

As a separate area of physics, electrodynamics consists of the following sections:

the doctrine of the field of motionless, but electrically charged physical bodies and particles;
the doctrine of the properties of electric current;
the doctrine of the interaction of the magnetic field and electromagnetic induction;
the doctrine of electromagnetic waves and oscillations.

All the above sections are combined into one whole by the theorem of D. Maxwell, who not only created and presented a coherent theory of the electromagnetic field, but also described all its properties, proving its real existence. The work of this particular scientist showed the scientific world that the electric and magnetic fields known at that time are just a manifestation of a single electromagnetic field that functions in different reference systems.

An essential part of physics is devoted to the study of electrodynamics and electromagnetic phenomena. This area largely claims the status of a separate science, since it not only investigates all the patterns of electromagnetic interactions, but also describes them in detail using mathematical formulas. Deep and long-term studies of electrodynamics have opened up new ways for the use of electromagnetic phenomena in practice, for the benefit of all mankind.

The relationship of magnetic induction B with the strength H of the magnetic field:

where μ is the magnetic permeability of an isotropic medium; μ 0 is the magnetic constant. In vacuum μ = 1, and then the magnetic induction in vacuum:

Biot-Savart-Laplace law: dB or dB=
dl,

where dB is the magnetic induction of the field created by a wire element of length dl with current I; r - radius - a vector directed from the conductor element to the point at which the magnetic induction is determined; α is the angle between the radius-vector and the direction of the current in the wire element.

Magnetic induction at the center of the circular current: V = ,

where R is the radius of the circular loop.

Magnetic induction on the axis of the circular current: B =
,

Where h is the distance from the center of the coil to the point at which the magnetic induction is determined.

Magnetic induction of the direct current field: V \u003d μμ 0 I / (2πr 0),

Where r 0 is the distance from the wire axis to the point at which the magnetic induction is determined.

Magnetic induction of the field created by a piece of wire with current (see Fig. 31, a and example 1)

B= (cosα 1 - cosα 2).

The designations are clear from the figure. The direction of the magnetic induction vector B is indicated by a dot - this means that B is directed perpendicular to the plane of the drawing towards us.

With a symmetrical arrangement of the ends of the wire relative to the point at which the magnetic induction is determined (Fig. 31 b), - сosα 2 = сosα 1 = сosα, then: B = cosα.

Solenoid field magnetic induction:

where n is the ratio of the number of turns of the solenoid to its length.

The force acting on a wire with current in a magnetic field (Ampère's law),

F = I , or F = IBlsinα,

Where l is the length of the wire; α is the angle between the direction of the current in the wire and the vector of magnetic induction B. This expression is valid for a uniform magnetic field and a straight piece of wire. If the field is not uniform and the wire is not straight, then Ampère's law can be applied to each element of the wire separately:

Magnetic moment of a flat circuit with current: p m \u003d n / S,

Where n is the unit vector of the normal (positive) to the contour plane; I is the strength of the current flowing through the circuit; S is the area of the contour.

Mechanical (rotational) moment acting on a current-carrying circuit placed in a uniform magnetic field,

M = , or M = p m B sinα,

Where α is the angle between vectors p m and B.

Potential energy (mechanical) of a circuit with current in a magnetic field: P mech = - p m B, or P mech = - p m B cosα.

The ratio of the magnetic moment p m to the mechanical L (momentum moment) of a charged particle moving in a circle orbit, =,

Where Q is the particle charge; m is the mass of the particle.

Lorentz force: F = Q , or F = Qυ B sinα ,

Where v is the speed of a charged particle; α is the angle between the vectors v and B.

Magnetic Flux:

A) in the case of a uniform magnetic field and a flat surface6

Ф = BScosα or Ф = B p S,

Where S is the contour area; α is the angle between the normal to the contour plane and the magnetic induction vector;

B) in the case of an inhomogeneous field and an arbitrary surface: Ф = V n dS

(integration is carried out over the entire surface).

Flux linkage (full flow): Ψ = NF.

This formula is true for a solenoid and a toroid with a uniform winding of N turns tightly adjacent to each other.

The work of moving a closed loop and in a magnetic field: A = IΔF.

EMF induction: ℰi = - .

Potential difference at the ends of a wire moving at a speed v in a magnetic field, U = Blυ sinα,

Where l is the length of the wire; α is the angle between the vectors v and B.

The charge flowing through a closed circuit when the magnetic flux penetrating this circuit changes:

Q = ΔФ/R, or Q = NΔФ/R = ΔΨ/R,

Where R is the loop resistance.

Loop inductance: L = F/I.

EMF of self-induction: ℰ s = - L .

Solenoid inductance: L = μμ 0 n 2 V,

Where n is the ratio of the number of turns of the solenoid to its length; V is the volume of the solenoid.

The instantaneous value of the current in a circuit with resistance R and inductance:

A) I = (1 - e - Rt \ L) (when the circuit is closed),

where ℰ is the EMF of the current source; t is the time elapsed after the circuit is closed;

B) I \u003d I 0 e - Rt \ L (when the circuit is opened), where I 0 is the current strength in the circuit at t \u003d 0; t is the time elapsed since the circuit was opened.

Magnetic field energy: W = .

Volumetric energy density of the magnetic field (the ratio of the energy of the magnetic field of the solenoid to its volume)

W \u003d VN / 2, or w \u003d B 2 / (2 μμ 0), or w \u003d μμ 0 H 2 /2,

Where B is the magnetic induction; H is the magnetic field strength.

Kinematic equation of harmonic oscillations of a material point: x = A cos (ωt + φ),

Where x is the offset; A is the amplitude of oscillations; ω is the angular or cyclic frequency; φ is the initial phase.

Acceleration rate of a material point making harmonic oscillations: υ = -Aω sin (ωt + φ); : υ \u003d -Aω 2 cos (ωt + φ);

Addition of harmonic oscillations of the same direction and the same frequency:

A) the amplitude of the resulting oscillation:

B) the initial phase of the resulting oscillation:

φ = arctan
.

The trajectory of a point participating in two mutually perpendicular oscillations: x = A 1 cos ωt; y \u003d A 2 cos (ωt + φ):

A) y = x, if the phase difference φ = 0;

B) y = - x, if the phase difference φ = ±π;

AT)
= 1 if phase difference φ = ± .

Plane traveling wave equation: y \u003d A cos ω (t - ),

Where y is the displacement of any of the points of the environment with the x coordinate at the moment t;

Υ is the speed of propagation of oscillations in the medium.

Relationship of the phase difference Δφ of oscillations with the distance Δx between the points of the medium, counted in the direction of propagation of the oscillations;

Δφ = Δx,

Where λ is the wavelength.

Examples of problem solving.

Example 1

A current 1 = 50 A flows along a straight wire segment 1 \u003d 80 cm long. Determine the magnetic induction B of the field created by this current at point A, equidistant from the ends of the wire segment and located at a distance r 0 \u003d 30 cm from its middle.

Decision.

To solve problems, we use the Biot-Savart-Laplace law and the principle of superposition of magnetic fields. The Biot-Savart-Laplace law will allow you to determine the magnetic induction dB created by the current element Idl. Note that the vector dB at point A is directed to the plane of the drawing. The principle of superposition allows one to use geometric summation 9 integration to determine B):

B = dB, (1)

Where the symbol l means that the integration extends over the entire length of the wire.

Let's write the Biot-Savart-Laplace law in vector form:

dB= ,

where dB is the magnetic induction created by a wire element of length dl with current I at a point determined by the radius-vector r; μ is the magnetic permeability of the medium in which the wire is located (in our case, μ = 1 *); μ 0 is the magnetic constant. Note that the dB vectors from different current elements are codirectional (Fig. 32), so expression (1) can be rewritten in scalar form: B = dB,

where dB = dl.

In the scalar expression of the Biot-Savart-Laplace law, the angle α is the angle between the current element Idl and the radius vector r. Thus:

B= dl. (2)

We transform the integrand so that there is one variable - the angle α. To do this, we express the length of the wire element dl through the angle dα: dl = rdα / sinα (Fig. 32).

Then the integrand dl can be written as:

= . Note that the variable r also depends on α, (r = r 0 /sin α); hence, =dα.

Thus, expression (2) can be rewritten as:

B = sinα dα.

Where α 1 and α 2 are the limits of integration.

AT Let's perform the integration: B = (cosα 1 – cosα 2). (3)

Note that with a symmetrical location of point A relative to a piece of wire cosα 2 = - cosα 1. With this in mind, formula (3) will take the form:

B = cosα 1 . (4)

From fig. 32 follows: cosα 1 =
=
.

Substituting the expressions cosα 1 into formula (4), we obtain:

B =
. (5)

Having made calculations using formula (5), we find: B = 26.7 μT.

The direction of the vector of magnetic induction B of the field created by direct current can be determined by the rule of the gimlet (the rule of the right screw). To do this, we draw a line of force (dashed line in Fig. 33) and draw vector B tangentially to it at the point of interest to us. The magnetic induction vector B at point A (Fig. 32) is directed perpendicular to the plane of the drawing from us.

R
is. 33, 34

Example 2

Two parallel endless long wires D and C, through which electric currents of strength I = 60 A flow in the same direction, are located at a distance d = 10 cm from each other. Determine the magnetic induction in the field created by conductors with current at point A (Fig. 34), separated from the axis of one conductor at a distance of r 1 \u003d 5 cm, from the other - r 2 \u003d 12 cm.

Decision.

To find the magnetic induction B at point A, we use the principle of superposition of magnetic fields. To do this, we determine the directions of the magnetic inductions B 1 and B 2 of the fields created by each conductor with current separately, and add them geometrically:

B \u003d B 1 + B 2.

The modulus of the vector B can be found using the cosine theorem:

B =
, (1)

Where α is the angle between the vectors B 1 and B 2.

Magnetic inductions B 1 and B 2 are expressed, respectively, in terms of current I and distances r 1 and r 2 from the wires to point A:

B 1 \u003d μ 0 I / (2πr 1); B 2 \u003d μ 0 I / (2πr 2).

Substituting the expressions B 1 and B 2 into formula (1) and taking μ 0 I / (2π) out of the sign of the root, we obtain:

B =
. (2)

Let's calculate cosα. Noting that α =
DAC (as angles with respectively perpendicular sides), by the cosine theorem we write:

d 2 = r +- 2r 1 r 2 cos α.

Where d is the distance between the wires. From here:

cos α =
; cos α =
= .

Let us substitute the numerical values of physical quantities into the formula (2) and perform the calculations:

B =

Tl \u003d 3.08 * 10 -4 Tl \u003d 308 μT.

Example 3

A current I = 80 A flows through a thin conducting ring with a radius R = 10 cm. Find the magnetic induction B at point A, equidistant from all points of the ring at a distance r = 20 cm.

Decision.

To solve the problem, we use the Biot-Savart-Laplace law:

dB=
,

where dB is the magnetic induction of the field created by the current element Idl at the point determined by the radius vector r.

We select an element dl on the ring and draw a radius vector r from it to point A (Fig. 35). Let's direct the dB vector in accordance with the gimlet rule.

According to the principle of superposition of magnetic fields, the magnetic induction At point A is determined by integration: B = dB,

Where integration is over all elements of the dl ring.

Let us decompose the dB vector into two components: dB , perpendicular to the plane of the ring, and dB ║ , parallel to the plane of the ring, i.e.

dB = dB + dB ║ .

t When: B = dB +dB║.

Noticing that dB ║ = 0 for symmetry reasons and that the vectors dB from different elements dl are co-directed, we replace the vector summation (integration) with a scalar one: B = dB ,

Where dB = dB cosβ and dB = dB = , (since dl is perpendicular to r and hence sinα = 1). Thus,

B= cosβ
dl=
.

After canceling by 2π and replacing cosβ with R/r (Fig. 35), we get:

B =
.

Let's check if the right side of the equation gives a unit of magnetic induction (T):

here we have used the defining formula for magnetic induction: B =
.

Then: 1Tl =
.

We express all quantities in SI units and perform calculations:

B =
Tl \u003d 6.28 * 10 -5 Tl, or B \u003d 62.8 μT.

Vector B is directed along the axis of the ring (dashed arrow in Fig. 35) in accordance with the rules of the gimlet.

Example 4

A long wire with current I = 50A is bent at an angle α = 2π/3. Determine the magnetic induction B at point A (36). Distance d = 5cm.

Decision.

A curved wire can be considered as two long wires, the ends of which are connected at point O (Fig. 37). In accordance with the principle of superposition of magnetic fields, the magnetic induction B at point A will be equal to the geometric sum of the magnetic inductions B 1 and B 2 of the fields created by segments of long wires 1 and 2, i.e. B \u003d B 1 + B 2. magnetic induction B 2 is zero. This follows from the Biot-Savart-Laplace law, according to which at points lying on the drive axis, dB = 0 ( = 0).

We find the magnetic induction B 1 using the relation (3) found in example 1:

B 1 = (cosα 1 - cosα 2),

G
de r 0 - the shortest distance from the wire l to point A

In our case, α 1 → 0 (the wire is long), α 2 = α = 2π/3 (cosα 2 = cos (2π/3) = -1/2). Distance r 0 \u003d d sin (π-α) \u003d d sin (π / 3) \u003d d
/2. Then the magnetic induction:

B 1 =
(1+1/2).

Since B \u003d B 1 (B 2 \u003d 0), then B \u003d
.

The vector B is co-directed with the vector B 1 is determined by the screw rule. On fig. 37 this direction is marked with a cross in a circle (perpendicular to the plane of the drawing, from us).

Checking the units is similar to that performed in example 3. Let's make the calculations:

B =
Tl \u003d 3.46 * 10 -5 Tl \u003d 34.6 μT.

Coulomb's law:

where F is the strength of the electrostatic interaction between two charged bodies;

q 1 , q 2 - electric charges of bodies;

ε is the relative, dielectric permittivity of the medium;

ε 0 \u003d 8.85 10 -12 F / m - electrical constant;

r is the distance between two charged bodies.

Linear charge density:

where d q- elementary charge per section of length d l.

Surface charge density:

where d q- elementary charge per surface d s.

Bulk charge density:

where d q- elementary charge, in volume d v.

Electric field strength:

where F – force acting on a charge q.

Gauss theorem:

where E is the strength of the electrostatic field;

d S – vector , the modulus of which is equal to the area of the penetrating surface, and the direction coincides with the direction of the normal to the site;

q is the algebraic sum of enclosed inside the surface d S charges.

Tension vector circulation theorem:

Electrostatic field potential:

where W p is the potential energy of a point charge q.

Point charge potential:

Field strength of a point charge:

The intensity of the field created by an infinite straight line of a uniformly charged line or an infinitely long cylinder:

where τ is the linear charge density;

r is the distance from the filament or the axis of the cylinder to the point where the field strength is determined.

The intensity of the field created by an infinite uniform charged plane:

where σ is the surface charge density.

Relationship of potential with tension in the general case:

E=- gradφ = .

Relationship between potential and strength in the case of a uniform field:

E= ,

where d– distance between points with potentials φ 1 and φ 2 .

Relationship between potential and strength in the case of a field with central or axial symmetry:

The work of the field forces to move the charge q from a point of the field with a potential φ 1 to the point of potential φ2:

A=q(φ 1 - φ 2).

Conductor capacitance:

where q is the charge of the conductor;

φ is the potential of the conductor, provided that at infinity the potential of the conductor is assumed to be zero.

Capacitor capacitance:

where q is the charge of the capacitor;

U is the potential difference between the capacitor plates.

Electric capacitance of a flat capacitor:

where ε is the permittivity of the dielectric located between the plates;

d is the distance between the plates;

S is the total area of the plates.

Capacitor battery capacity:

b) with parallel connection:

Energy of a charged capacitor:

where q is the charge of the capacitor;

U is the potential difference between the plates;

C is the capacitance of the capacitor.

DC power:

where d q- the charge flowing through the cross section of the conductor during the time d t.

current density:

where I- current strength in the conductor;

S is the area of the conductor.

Ohm's law for a circuit section that does not contain EMF:

where I- current strength in the area;

R- section resistance.

Ohm's law for a circuit section containing EMF:

where I- current strength in the area;

U- voltage at the ends of the section;

R- the total resistance of the section;

ε – source emf.

Ohm's law for a closed (complete) circuit:

where I- current strength in the circuit;

R- external resistance of the circuit;

r is the internal resistance of the source;

ε – source emf.

Kirchhoff's laws:

2. ,

where is the algebraic sum of the strengths of the currents converging in the node;

- algebraic sum of voltage drops in the circuit;

is the algebraic sum of the EMF in the circuit.

Conductor resistance:

where R– conductor resistance;

ρ is the resistivity of the conductor;

l- conductor length;

Conductor conductivity:

where G is the conductivity of the conductor;

γ is the specific conductivity of the conductor;

l- conductor length;

S is the cross-sectional area of the conductor.

Conductor system resistance:

a) in series connection:

a) in parallel connection:

Current work:

where A– current work;

U- voltage;

I– current strength;

R- resistance;

t- time.

Current power:

Joule–Lenz law

where Q is the amount of heat released.

Ohm's law in differential form:

j=γ E ,

where j is the current density;

γ – specific conductivity;

E is the electric field strength.

Relationship of magnetic induction with magnetic field strength:

B=μμ 0 H ,

where B is the magnetic induction vector;

μ is the magnetic permeability;

H is the strength of the magnetic field.

Biot-Savart-Laplace law:

where d B is the induction of the magnetic field created by the conductor at some point;

μ is the magnetic permeability;

μ 0 \u003d 4π 10 -7 H / m - magnetic constant;

I- current strength in the conductor;

d l – conductor element;

r is the radius vector drawn from the element d l conductor to the point where the magnetic field induction is determined.

The total current law for a magnetic field (theorem of the circulation of the vector B):

where n- the number of conductors with currents covered by the circuit L arbitrary shape.

Magnetic induction at the center of the circular current:

where R is the radius of the circle.

Magnetic induction on the axis of circular current:

where h is the distance from the center of the coil to the point at which the magnetic induction is determined.

Magnetic induction of direct current field:

where r 0 is the distance from the wire axis to the point where the magnetic induction is determined.

Solenoid field magnetic induction:

B=μμ 0 ni,

where n is the ratio of the number of turns of the solenoid to its length.

Amp power:

d F =I,

where d F – Ampere power;

I- current strength in the conductor;

d l - conductor length;

B– magnetic field induction.

Lorentz force:

F=q E +q[v B ],

where F is the Lorentz force;

q is the particle charge;

E is the electric field strength;

v is the speed of the particle;

B– magnetic field induction.

Magnetic Flux:

a) in the case of a uniform magnetic field and a flat surface:

Φ=B n S,

where Φ – magnetic flux;

B n is the projection of the magnetic induction vector onto the normal vector;

S is the contour area;

b) in the case of an inhomogeneous magnetic field and an arbitrary projection:

Flux linkage (full flow) for toroid and solenoid:

where Ψ – full flow;

N is the number of turns;

Φ - magnetic flux penetrating one turn.

Loop inductance:

Solenoid inductance:

L=μμ 0 n 2 V,

where L is the inductance of the solenoid;

μ is the magnetic permeability;

μ 0 is the magnetic constant;

n is the ratio of the number of turns to its length;

V is the volume of the solenoid.

Faraday's law of electromagnetic induction:

where ε i– EMF of induction;

– change in total flow per unit time.

The work of moving a closed loop in a magnetic field:

A=IΔ Φ,

where A- work on moving the contour;

I- current strength in the circuit;

Δ Φ – change in the magnetic flux penetrating the circuit.

EMF of self-induction:

Magnetic field energy:

Volumetric energy density of the magnetic field:

where ω is the volumetric energy density of the magnetic field;

B– magnetic field induction;

H– magnetic field strength;

μ is the magnetic permeability;

μ 0 is the magnetic constant.

3.2. Concepts and definitions

? List the properties of an electric charge.

1. There are two types of charges - positive and negative.

2. Charges of the same name repel, unlike charges attract.

3. Charges have the property of discreteness - all are multiples of the smallest elementary.

4. The charge is invariant, its value does not depend on the frame of reference.

5. The charge is additive - the charge of the system of bodies is equal to the sum of the charges of all the bodies of the system.

6. The total electric charge of a closed system is a constant value

7. A stationary charge is a source of an electric field, a moving charge is a source of a magnetic field.

? Formulate Coulomb's law.

The force of interaction between two fixed point charges is proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. The force is directed along the line connecting the charges.

? What is an electric field? Electric field strength? Formulate the principle of superposition of electric field strength.

An electric field is a type of matter associated with electric charges and transmitting the action of one charge to another. Tension - the power characteristic of the field, equal to the force acting on a unit positive charge placed at a given point in the field. The principle of superposition - the field strength created by a system of point charges is equal to the vector sum of the field strengths of each charge.

? What is called the lines of force of the electrostatic field? List the properties of lines of force.

The line, the tangent at each point of which coincides with the direction of the field strength vector, is called the force line. Properties of lines of force - start on positive, end on negative charges, do not interrupt, do not intersect with each other.

? Define an electric dipole. dipole field.

A system of two equal in absolute value, opposite in sign, point electric charges, the distance between which is small compared to the distance to the points where the action of these charges is observed. The intensity vector has a direction opposite to the electric moment vector of the dipole (which, in turn, is directed from negative charge to positive).

? What is the potential of an electrostatic field? Formulate the principle of potential superposition.

A scalar quantity numerically equal to the ratio of the potential energy of an electric charge placed at a given point in the field to the magnitude of this charge. The principle of superposition - the potential of a system of point charges at a certain point in space is equal to the algebraic sum of the potentials that these charges would create separately at the same point in space.

? What is the relationship between tension and potential?

E=- (E - field strength at a given point of the field, j - potential at this point.)

? Define the concept of "flux of the electric field strength vector". Formulate the electrostatic theorem of Gauss.

For an arbitrary closed surface, the intensity vector flux E electric field F E= . Gauss theorem:

= (here Q i are charges covered by a closed surface). Valid for a closed surface of any shape.

? What substances are called conductors? How are charges and electrostatic field distributed in a conductor? What is electrostatic induction?

Conductors are substances in which, under the influence of an electric field, free charges can move in an orderly manner. Under the action of an external field, the charges are redistributed, creating their own field, equal in absolute value to the external one and directed oppositely. Therefore, the resulting tension inside the conductor is 0.

Electrostatic induction is a type of electrization in which, under the action of an external electric field, the redistribution of charges between parts of a given body occurs.

? What is the electric capacitance of a solitary conductor, a capacitor. How to determine the capacitance of a flat capacitor, a bank of capacitors connected in series, in parallel? Unit of measure for electrical capacity.

Solitary conductor: where With-capacity, q- charge, j - potential. The unit of measure is farad [F]. (1 F is the capacitance of the conductor, in which the potential increases by 1 V when a charge of 1 C is imparted to the conductor).

Capacitance of a flat capacitor. Serial connection: . Parallel connection: C total = C 1 +C 2 +…+С n

? What substances are called dielectrics? What types of dielectrics do you know? What is dielectric polarization?

Dielectrics are substances in which, under normal conditions, there are no free electric charges. There are dielectrics polar, non-polar, ferroelectric. Polarization is the process of orientation of dipoles under the influence of an external electric field.

? What is an electrical displacement vector? Formulate Maxwell's postulate.

Electrical displacement vector D characterizes the electrostatic field created by free charges (i.e. in vacuum), but with such a distribution in space, which is available in the presence of a dielectric. Maxwell's postulate: . Physical meaning - expresses the law of creating electric fields by the action of charges in arbitrary media.

? Formulate and explain the boundary conditions for the electrostatic field.

When the electric field passes through the interface between two dielectric media, the intensity and displacement vectors change abruptly in magnitude and direction. The relations characterizing these changes are called boundary conditions. There are 4 of them:

(3), (4)

? How is the energy of an electrostatic field determined? Energy density?

Energy W= ( E- field strength, e-dielectric constant, e 0 - electrical constant, V- field volume), energy density

? Define the concept of "electric current". Types of currents. Characteristics of electric current. What condition is necessary for its occurrence and existence?

Current is the ordered movement of charged particles. Types - conduction current, ordered movement of free charges in a conductor, convection - occurs when a charged macroscopic body moves in space. For the emergence and existence of a current, it is necessary to have charged particles capable of moving in an orderly manner, and the presence of an electric field, the energy of which, being replenished, would be spent on this ordered movement.

? Give and explain the continuity equation. Formulate the condition of current stationarity in integral and differential forms.

Continuity equation. Expresses in differential form the law of conservation of charge. The condition of stationarity (constancy) of the current in integral form: and differential -.

? Write down Ohm's law in integral and differential forms.

Integral form - ( I-current, U- voltage, R-resistance). Differential form - ( j - current density, g - electrical conductivity, E - field strength in the conductor).

? What are third party forces? EMF?

External forces separate charges into positive and negative. EMF - the ratio of work to move the charge along the entire closed circuit to its value

? How is work and power determined?

When moving charge q through an electrical circuit at the ends of which voltage is applied U, electric field does work , current power (t-time)

? Formulate Kirchhoff's rules for branched chains. What conservation laws are incorporated in Kirchhoff's rules? How many independent equations should be composed on the basis of the first and second Kirchhoff laws?

1. The algebraic sum of the currents converging in the node is 0.

2. In any arbitrarily chosen closed circuit, the algebraic sum of the voltage drops is equal to the algebraic sum of the EMF occurring in this circuit. Kirchhoff's first rule follows from the law of conservation of electric charge. The number of equations in the sum should be equal to the number of sought values (all resistances and EMF should be included in the system of equations).

? Electric current in gas. Processes of ionization and recombination. The concept of plasma.

Electric current in gases is the directed movement of free electrons and ions. Under normal conditions, gases are dielectrics, they become conductors after ionization. Ionization is the process of forming ions by separating electrons from gas molecules. Occurs due to the influence of an external ionizer - strong heating, X-ray or ultraviolet radiation, electron bombardment. Recombination is a process that is the reverse of ionization. Plasma is a fully or partially ionized gas in which the concentrations of positive and negative charges are equal.

? Electric current in vacuum. Thermionic emission.

Current carriers in vacuum are electrons emitted due to emission from the surface of the electrodes. Thermionic emission is the emission of electrons by heated metals.

? What do you know about the phenomenon of superconductivity?

The phenomenon in which the resistance of some pure metals (tin, lead, aluminum) drops to zero at temperatures close to absolute zero.

? What do you know about the electrical resistance of conductors? What is resistivity, its dependence on temperature, electrical conductivity? What do you know about series and parallel connection of conductors. What is a shunt, additional resistance?

Resistance - a value directly proportional to the length of the conductor l and inversely proportional to the area S cross-section of the conductor: (r-specific resistance). Conductivity is the reciprocal of resistance. Resistivity (resistance of a conductor 1 m long with a cross section of 1 m 2). Resistivity is temperature dependent, where a is the temperature coefficient, R and R 0 , r and r 0 are resistances and specific resistances at t and 0 0 С. Parallel - , sequential R=R 1 +R 2 +…+R n. A shunt is a resistor connected in parallel with an electrical measuring instrument to divert part of the electric current in order to expand the measurement limits.

? A magnetic field. What sources can create a magnetic field?

A magnetic field is a special kind of matter through which moving electric charges interact. The reason for the existence of a constant magnetic field is a fixed conductor with a constant electric current, or permanent magnets.

? Formulate Ampère's law. How do conductors interact in which current flows in one (opposite) direction?

Ampere's force is acting on a current-carrying conductor.

B - magnetic induction, I- conductor current, D l is the length of the conductor section, a is the angle between the magnetic induction and the conductor section. In one direction they attract, in the opposite direction they repel.

? Define the ampere force. How to determine its direction?

This is the force acting on a current-carrying conductor placed in a magnetic field. We define the direction as follows: we position the palm of the left hand so that it includes the lines of magnetic induction, and four outstretched fingers are directed along the current in the conductor. The bent thumb will show the direction of Ampere's force.

? Explain the movement of charged particles in a magnetic field. What is the Lorentz force? What is its direction?

A moving charged particle creates its own magnetic field. If it is placed in an external magnetic field, then the interaction of the fields will manifest itself in the emergence of a force acting on the particle from the external field - the Lorentz force. Direction - according to the rule of the left hand. For positive charge - vector B enters the palm of the left hand, four fingers are directed along the movement of the positive charge (velocity vector), the bent thumb shows the direction of the Lorentz force. On a negative charge, the same force acts in the opposite direction.

(q-charge, v-speed, B- induction, a - angle between the direction of velocity and magnetic induction).

? Frame with current in a uniform magnetic field. How is magnetic moment determined?

The magnetic field has an orienting effect on the frame with current, turning it in a certain way. The torque is given by: M =p m x B , where p m- the vector of the magnetic moment of the loop with current, equal to IS n (current per contour surface area, per unit normal to the contour), B - vector of magnetic induction, quantitative characteristic of the magnetic field.

? What is the magnetic induction vector? How to determine its direction? How is a magnetic field shown graphically?

The magnetic induction vector is the power characteristic of the magnetic field. The magnetic field is visualized using lines of force. At each point of the field, the tangent to the field line coincides with the direction of the magnetic induction vector.

? Formulate and explain the Biot-Savart-Laplace law.

The Biot-Savart-Laplace law allows you to calculate for a current-carrying conductor I magnetic induction of the field d B , created at an arbitrary point of the field d l conductor: (here m 0 is the magnetic constant, m is the magnetic permeability of the medium). The direction of the induction vector is determined by the rule of the right screw, if the translational movement of the screw corresponds to the direction of the current in the element.

? Formulate the principle of superposition for a magnetic field.

Superposition principle - the magnetic induction of the resulting field created by several currents or moving charges is equal to the vector sum of the magnetic inductions of the added fields created by each current or moving charge separately:

? Explain the main characteristics of a magnetic field: magnetic flux, magnetic field circulation, magnetic induction.

magnetic flux F through any surface S call the value equal to the product of the modulus of the magnetic induction vector and the area S and the cosine of the angle a between the vectors B and n (outer normal to the surface). Vector circulation B along a given closed contour is called an integral of the form , where d l - vector of elementary contour length. Vector circulation theorem B : vector circulation B along an arbitrary closed circuit is equal to the product of the magnetic constant and the algebraic sum of the currents covered by this circuit. The magnetic induction vector is the power characteristic of the magnetic field. The magnetic field is visualized using lines of force. At each point of the field, the tangent to the field line coincides with the direction of the magnetic induction vector.

? Write down and comment on the condition of solenoidality of the magnetic field in integral and differential forms.

Vector fields in which there are no sources and sinks are called solenoidal. The condition of solenoidality of the magnetic field in integral form: and differential form:

? Magnetics. Types of magnets. Feromagnets and their properties. What is hysteresis?

A substance is magnetic if it is capable of acquiring a magnetic moment (be magnetized) under the action of a magnetic field. Substances that are magnetized in an external magnetic field against the direction of the field are called diamagnets. Those that are magnetized in an external magnetic field in the direction of the field are called paramagnets. These two classes are called weakly magnetic substances. Strongly magnetic substances that are magnetized even in the absence of an external magnetic field are called ferromagnets. . Magnetic hysteresis - the difference in the values of the magnetization of a ferromagnet at the same intensity H of the magnetizing field, depending on the value of the preliminary magnetization. Such a graphical dependence is called a hysteresis loop.

? Formulate and explain the law of total current in integral and differential forms (basic equations of magnetostatics in matter).

? What is electromagnetic induction? Formulate and explain the basic law of electromagnetic induction (Faraday's law). Formulate Lenz's rule.

The phenomenon of the occurrence of an electromotive force (EMF of induction) in a conductor located in an alternating magnetic field or moving in a constant in a constant magnetic field is called electromagnetic induction. Faraday's law: whatever the reason for the change in the flux of magnetic induction, covered by a closed conducting circuit, that occurs in the EMF circuit

The minus sign is determined by the Lenz rule - the induction current in the circuit always has such a direction that the magnetic field it creates prevents a change in the magnetic flux that caused this induction current.

? What is the phenomenon of self-induction? What is inductance, units of measurement? Currents during the closing and opening of the electrical circuit.

The occurrence of induction EMF in a conducting circuit under the influence of its own magnetic field when it changes, which occurs as a result of a change in the current strength in the conductor. Inductance is a proportionality factor depending on the shape and dimensions of the conductor or circuit, [H]. In accordance with the Lenz rule, the EMF of self-induction prevents the increase in current strength when the circuit is turned on and the decrease in current strength when the circuit is turned off. Therefore, the magnitude of the current strength cannot change instantly (the mechanical analogue is inertia).

? The phenomenon of mutual induction. Mutual induction coefficient.

If two fixed circuits are located close to each other, then when the current strength in one circuit changes, an emf occurs in the other circuit. This phenomenon is called mutual induction. Proportionality coefficients L 21 and L 12 is called the mutual inductance of the circuits, they are equal.

? Write Maxwell's equations in integral form. Explain their physical meaning.

; ;

; .

It follows from Maxwell's theory that the electric and magnetic fields cannot be considered as independent - a change in time of one leads to a change in the other.

? The energy of the magnetic field. Magnetic field energy density.

Energy, L-inductance, I- current strength.

Density , AT- magnetic induction, H is the magnetic field strength, V-volume.

? The principle of relativity in electrodynamics

The general laws of electromagnetic fields are described by Maxwell's equations. In relativistic electrodynamics, it is established that the relativistic invariance of these equations takes place only under the condition of relativity of electric and magnetic fields, i.e. when the characteristics of these fields depend on the choice of inertial frames of reference. In a moving system, the electric field is the same as in a stationary system, but in a moving system there is a magnetic field, which is not present in a stationary system.

Vibrations and waves