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physics
electricity and magnetism

Questions and Answers of Electricity and Magnetism

The magnetic induction in vacuum at a plane surface of a magnetic is equal to B and the vector B forms an angle θ with the normal n of the surface (Fig. 3.75).The permeability of the magnetic
A direct current I flows in a long round uniform cylindrical wire made of paramagnetic with susceptibility X Find: (a) The surface molecular current Is; (b) The volume molecular current Iv. How
Half of an infinitely long straight current-carrying solenoid is filled with magnetic substance as shown in Fig. 3.75. Draw the approximate plots of magnetic induction B, strength H, and
An infinitely long wire with a current I flowing in it is located in the boundary plane between two non-conducting media with permeabilities μ1 and μ2. Find the modulus of the magnetic
A round current-carrying loop lies in the plane boundary between magnetic and vacuum. The permeability of the magnetic is equal to μ. Find the magnetic induction B at an arbitrary point on the
When a ball made of uniform magnetic is introduced into an external uniform magnetic field with induction B0, it gets uniformly magnetized. Find the magnetic induction B inside the ball with
N = 300 turns of thin wire are uniformly wound on a permanent magnet shaped as a cylinder whose length is equal to l = 15 cm. When a current I = 3.0 A was passed through the wiring the field outside
A permanent magnet is shaped as a ring with a narrow gap between the poles. The mean diameter of the ring equals d = 20 cm. The width of the gap is equal to b = 2.0 mm and the magnetic induction in
An iron core shaped as a tore with mean radius R = 250 mm supports a winding with the total number of turns N = 1000. The core has a cross-cut of width b = 1.00 ram. With a current I = 0.85 A flowing
Fig. 3.76 illustrates a basic magnetization curve of iron (commercial purity grade). Using this plot, draw the permeability μ as a function of the magnetic field strength H. At what value of H is
A thin iron ring with mean diameter d = 50 cm supports a winding consisting of N = 800 turns carrying current I = 3.0 A. The ring has a cross-cut of width b = 2.0 mm. Neglecting the scattering of the
A long thin cylindrical rod made of paramagnetic with magnetic susceptibility % and having a cross-sectional area S is located along the axis of a current-carrying coil. One end of the rod is located
In the arrangement shown in Fig. 3.77 it is possible to measure (by means of a balance) the force with which a paramagnetic ball of volume V = 41 mm3 is attrabted to a pole of the electromagnet net
A small ball of volume V made of paramagnetic with susceptibility X was slowly displaced along the axis of a current-carrying coil from the point where the magnetic induction equals B out to the
A wire bent as a parabola y = ax2 is located in a uniform magnetic field of induction B, the vector B being perpendicular to the plane x, y. At the moment t = 0 a connector starts sliding translation
A rectangular loop with a sliding connector of length l is located in a uniform magnetic field perpendicular to the loop plane (Fig. 3.79). The magnetic induction is equal to B. The connector has an
A metal disc of radius a = 25 cm rotates with a constant angular velocity w = 130 rad/s about its axis. Find the potential difference between the centre and the rim of the disc if (a) The external
A thin wire AC shaped as a semi-circle of diameter d = 20cm rotates with a constant angular velocity w = 100 rad/s in a uniform magnetic field of induction B= 5.0 mT, with w ↑↑ B. The
A wire loop enclosing a semi-circle of radius a is located on the boundary of a uniform magnetic field of induction B (Fig. 3.80). At the moment t = 0 the loop is set into rotation with a constant
A long straight wire carrying a current I and a H-shaped conductor with sliding connector are located in the same plane as shown in Fig. 3.81. The connector of length l and resistance R slides to the
A square frame with side a and a long straight wire carrying a current I are located in the same plane as shown in Fig. 3.82. The frame translates to the right with a constant velocity v. Find the
A metal rod of mass m can rotate about a horizontal axis O, sliding along a circular conductor of radius a (Fig. 3.83). The arrangement is located in a uniform magnetic field of induction B directed
A copper connector of mass m slides down two smooth copper bars, set at an angle a to the horizontal, due to gravity (Fig. 3.84). At the top the bars are interconnected through a resistance R. The
The system differs from the one examined in the foregoing problem (Fig. 3.84) by a capacitor of capacitance C replacing the resistance R. Find the acceleration of the connector.
A wire shaped as a semi-circle of radius a rotates about an axis 00' with an angular velocity w in a uniform magnetic field of induction B (Fig. 3.85). The rotation axis is perpendicular to the field
A small coil is introduced between the poles of an electromagnet so that its axis coincides with the magnetic field direction. The cross-sectional area of the coil is equal to S = 3.0 mm 2, the
A square wire frame with side a and a straight conductor carrying a constant current I are located in the same plane (Fig. 3.86). The inductance and the resistance of the frame are equal to L and R
A long straight wire carries a current I0. At distances a and b from it there are two other wires, parallel to the former one, which are interconnected by a resistance R (Fig. 3.87). A connector
A conducting rod AB of mass m slides without friction over two long conducting rails separated by a distance l (Fig. 3.88). At the left end the rails are interconnected by a resistance R. The system
A connector AB can slide without friction along a H-shaped conductor located in a horizontal plane (Fig. 3.89). The connector has a length l, mass m, and resistance R. The whole system is located in
Fig. 3.90 illustrates plane figures made of thin conductors which are located in a uniform magnetic field directed away from a reader beyond the plane of the drawing. The magnetic induction starts
A plane loop shown in Fig. 3.91 is shaped as two squares with sides a = 20 cm and b = 10 cm and is introduced into a uniform magnetic field at right angles to the loop's plane. The magnetic induction
A plane spiral with a great number N of turns wound tightly to one another is located in a uniform magnetic field perpendicular to the spiral's plane. The outside radius of the spiral's turns is
A H-shaped conductor is located in a uniform magnetic field perpendicular to the plane of the conductor and varying with time at the rate B = 0.10 T/s. A conducting connector starts moving with an
In a long straight solenoid with cross-sectional radius a and number of turns per unit length n a current varies with a constant velocity i A/s. Find the magnitude of the eddy current field strength
A long straight solenoid of cross-sectional diameter d = 5 cm and with n = 20 turns per one cm of its length has a round turn of copper wire of cross-sectional area S = 1.0 mm2 tightly put on its
A long straight solenoid of cross-sectional diameter d = 5 cm and with n = 20 turns per one cm of its length has a round turn of copper wire of cross-sectional area S = 1.0 mm2 tightly put on its
A thin non-conducting ring of mass m carrying a charge q can freely rotate about its axis. At the initial moment the ring was at rest and no magnetic field was present. Then a practically uniform
A thin wire ring of radius a and resistance r is located inside a long solenoid so that their axes coincide. The length of the solenoid is equal to l, its cross-sectional radius, to b. At a certain
A magnetic flux through a stationary loop with a resistance R varies during the time interval τ as Ф = at (τ – t). Find the amount of heat generated in the loop during that time.
In the middle of a long solenoid there is a coaxial ring of square cross-section, made of conducting material with resistivity p. The thickness of the ring is equal to h, its inside and outside radii
How many meters of a thin wire are required to manufacture a solenoid of length 10 = 100 cm and inductance L = 1.0 mH if the solenoid's cross-sectional diameter is considerably less than its length?
Find the inductance of a solenoid of length l whose winding is made of copper wire of mass m. The winding resistance is equal to R. The solenoid diameter is considerably less than its length.
A coil of inductance L = 300 mH and resistance R =140 mΩ is connected to a constant voltage source. How soon will the coil current reach η = 50% of the steady-state value?
Calculate the time constant τ of a straight solenoid of length l = 1.0 m having a single-layer winding of copper wire whose total mass is equal to m =1.0 kg. The cross-sectional diameter of the
Find the inductance of a unit length of a cable consisting of two thin-walled coaxial metallic cylinders if the radius of the out- side cylinder is η = 3.6 times that of the inside one. The
Calculate the inductance of a doughnut solenoid whose inside radius is equal to b and cross-section has the form of a square with side a. The solenoid winding consists of N turns. The space inside
Calculate the inductance of a unit length of a double tape line (Fig. 3.92) if the tapes are separated by a distance h which is considerably less than their width b, namely, b/h = 50.
Find the inductance of a unit length of a double line if the radius of each wire is η times less than the distance between the axes of the wires. The field inside the wires is to be neglected,
A superconducting round ring of radius a and inductance L was located in a uniform magnetic field of induction B. The ring plane was parallel to the vector B, and the current in the ring was equal to
A current I0 = 1.9 A flows in a long closed solenoid. The wire it is wound of is in a superconducting state. Find the current flowing in the solenoid when the length of the solenoid is increased by
A ring of radius a = 50 mm made of thin wire of radius b = 1.0 mm was located in a uniform magnetic field with induction B = 0.50 mT so that the ring plane was perpendicular to the vector B. Then the
A closed circuit consists of a source of constant emf ε and a choke coil of inductance L connected in series. The active resistance of the whole circuit is equal to R. At the moment t = 0 the
Find the time dependence of the current flowing through the inductance L of the circuit shown in Fig. 3.93 after the switch Sw is shorted at the moment t = 0.
In the circuit shown in Fig. 3.94 an emf $, a resistance R, and coil inductances L1 and L2 are known. The internal resistance of the source and the coil resistances are negligible. Find the steady-
Calculate the mutual inductance of a long straight wire and a rectangular frame with sides a and b. The frame and the wire lie in the same plane, with the side b being closest to the wire, separated
Determine the mutual inductance of a doughnut coil and an infinite straight wire passing along its axis. The coil has a rectangular cross-section, its inside radius is equal to a and the outside one,
Two thin concentric wires shaped as circles with radii a and b lie in the same plane. Allowing for a
A small cylindrical magnet M (Fig. 3.95) is placed in the centre of a thin coil of radius a consisting of N turns. The coil is connected to a ballistic galvanometer. The active resistance of the
Find the approximate formula expressing the mutual inductance of two thin coaxial loops of the same radius a if their centers are separated by a distance l, with l: >> a.
There are two stationary loops with mutual inductance Ltd. The current in one of the loops starts to be varied as I1 = at, where a is a constant, t is time. Find the time dependence 12 (t) of the
A coil of inductance L = 2.0μH and resistance R = 1.0Ω is connected to a source of constant emf ε = 3.0 V (Fig. 3.96). A resistance R0 = 2.0Ω is connected in parallel with the
An iron tore supports N = 500 turns. Find the magnetic field energy if a current I = 2.0 A produces a magnetic flux across the tore’s cross-section equal to Ф = 1.0mWb.
An iron core shaped as a doughnut with round cross-section of radius a = 3.0 cm carries a winding of N = 1000 turns through which a current I = 1.0 A flows. he mean radius of the doughnut is b = 32
A thin ring made of a magnetic has a mean diameter d = 30 cm and supports a winding of N = 800 turns. The cross-sectional area of the ring is equal to S = 5.0 cm e. The ring has a cross-cut of width
A long cylinder of radius a carrying a uniform surface charge rotates about its axis with an angular velocity θ. Find the magnetic field energy per unit length of the cylinder if the linear
At what magnitude of the electric field strength in vacuum the volume energy density of this field is the same as that of the magnetic field with induction B = 1.0 T (also in vacuum).
A thin uniformly charged ring of radius a = 10 cm rotates about its axis with an angular velocity w = 100 rad/s. Find the ratio of volume energy densities of magnetic and electric fields on the axis
Using the expression for volume density of magnetic energy, demonstrate that the amount of work contributed to magnetization of a unit volume of Para-or diamagnetic, is equal to A = – JB/2.
Two identical coils, each of inductance L, are interconnected (a) In series, (b) In parallel. Assuming the mutual inductance of the coils to be negligible, find the inductance of the system in both
Two solenoids of equal length and almost equal cross-sectional area are fully inserted into one another. Find their mutual inductance if their inductances are equal to L1 and L2.
Demonstrate that the magnetic energy of interaction of two current-carrying loops located in vacuum can be represented as Wia = (1/μ0) ∫ B1B2 dV, where B1 and B2 are the magnetic
Find the interaction energy of two loops carrying currents I1 and 12 if both loops are shaped as circles of radii a and b, with a
The space between two concentric metallic spheres is filled up with a uniform poorly conducting medium of resistivity p and permittivity e. At the moment t = 0 the inside sphere obtains a certain
A parallel-plate capacitor is formed by two discs with a uniform poorly conducting medium between them. The capacitor was initially charged and then disconnected from a voltage source. Neglecting the
A parallel-plate air condenser whose each plate has an area S = 100 cm2 is connected in series to an ac circuit. Find the electric field strength amplitude in the capacitor if the sinusoidal current
The space between the electrodes of a parallel-plate capacitor is filled with a uniform poorly conducting medium of conductivity σ and permittivity ε. The capacitor plates shaped as round
A long straight solenoid has n turns per unit length. An alternating current I = I m sin cot flows through it. Find the displacement current density as a function of the distance r from the solenoid
A point charge q moves with a non-relativistic velocity v = const. Find the displacement current density j d at a point located at a distance r from the charge on a straight line (a) Coinciding with
A thin wire ring of radius a carrying a charge q approaches the observation point P so that its centre moves rectilinearly with a constant velocity v. The plane of the ring remains perpendicular to
A point charge q moves with a non-relativistic velocity v = const. Applying the theorem for the circulation of the vector I4 around the dotted circle shown in Fig. 3.97, find H at the point A as a
Using Maxwell's equations, show that (a) A time-dependent magnetic field cannot exist without an electric field; (b) A uniform electric field cannot exist in the presence of a time-dependent magnetic
Demonstrate that the law of electric charge conservation, i.e. ∆ ∙.j =- ∂p/∂t, follows from Maxwell's equations.
Demonstrate that Maxwell's equations ∆ × E =- ∂B/∂t and ∆ x B = 0 are compatible, i.e. the first one does not contradict the second one.
In a certain region of the inertial reference frame there is magnetic field with induction B rotating with angular velocity to. Find V x E in this region as a function of vectors to and B.
In the inertial reference frame K there is a uniform magnetic field with induction B. Find the electric field strength in the frame K' which moves relative to the frame K with a non-relativistic
A large plate of non-ferromagnetic material moves with a constant velocity v = 90 cm/s in a uniform magnetic field with induction B = 50 mT as shown in Fig. 3.98. Find the surface density of electric
A long solid aluminum cylinder of radius a = 5.0 cm rotates about its axis in a uniform magnetic field with induction B = 10 mT. The angular velocity of rotation equals w = 45 rad/s, with w
A non-relativistic point charge q moves with a constant velocity v. Using the field transformation formulas, find the magnetic induction B produced by this charge at the point whose position relative
Using Eqs (3.6h), demonstrate that if in the inertial reference frame K there is only electric or only magnetic field, in any other inertial frame K' both electric and magnetic fields will coexist
In an inertial reference frame K there is only magnetic field with induction B = b (yi – xj)/ (x2 + y2), where b is a constant, i and j are the unit vectors of the x and y axes. Find the electric
In an inertial reference frame K there is only electric field of strength E = a (xi + yj)/(x2 + y2), where a is a constant, i and j are the unit vectors of the x and y axes. Find the magnetic
Demonstrate that the" transformation formulas (3.6h) follow from the formulas (3.6i) at v 0
In an inertial reference frame K there is only a uniform electric field E = 8 kV/m in strength. Find the modulus and direction (a) Of the vector E', (b) Of the vector B' in the inertial reference
Solve a problem differing from the foregoing one by a magnetic field with induction B = 0.8 T replacing the electric field.
Electromagnetic field has two invariant quantities. Using the transformation formulas (3.6i), demonstrate that these quantities are (a) EB; (b) E2 – c2B2.
In an inertial reference frame K there are two uniform mutually perpendicular fields: an electric field of strength E = 40 kV/m and a magnetic field induction B = 0.20 mT. Find the electric strength
A point charge q moves uniformly and rectilinearly with a relativistic velocity equal to a β fraction of the velocity of light (β = v/c). Find the electric field strength E produced by the
At the moment t = 0 an electron leaves one plate of a parallel-plate capacitor with a negligible velocity. An accelerating voltage, varying as V = at, where a = 100 V/s, is applied between the
A proton accelerated by a potential difference V gets into the uniform electric field of a parallel-plate capacitor whose plates extend over a length l in the motion direction. The field strength

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