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physics
electricity and magnetism
College Physics 7th edition Jerry D. Wilson, Anthony J. Buffa, Bo Lou - Solutions
A particle with specific charge q/m moves rectilinearly due to an electric field E = Eo = ax, where a is a positive constant, x is the distance from the point where the particle was initially at rest. Find: (a) The distance covered by the particle till the moment it came to a standstill; (b) The
An electron starts moving in a uniform electric field of strength E = 10 kV/cm. How soon after the start will the kinetic energy of the electron become equal to its rest energy?
Determine the acceleration of a relativistic electron moving along a uniform electric field of strength E at the moment when its kinetic energy becomes equal to T.
At the moment t = 0 a relativistic proton flies with a velocity vo into the region where there is a uniform transverse electric field of strength E, with vo ┴ E. Find the time dependence of? (a) The angle 0 between the proton's velocity vector v and the initial direction of its motion;
A proton accelerated by a potential difference V = 500 kV flies through a uniform transverse magnetic field with induction B = 0.51 T. The field occupies a region of space d = 10 cm in thickness (Fig. 3.99). Find the angle a through which the proton deviates from the initial direction of its motion.
A charged particle moves along a circle of radius r = 100 mm in a uniform magnetic field with induction B = 10.0 mT. Find its velocity and period of revolution if that particle is (a) A non-relativistic proton; (b) A relativistic electron.
A relativistic particle with charge q and rest mass mo moves along a circle of radius r in a uniform magnetic field of induction B. Find: (a) The modulus of the particle's momentum vector; (b) The kinetic energy of the particle; (c) The acceleration of the particle.
Up to what values of kinetic energy does the period of revolution of an electron and a proton in a uniform magnetic field exceed that at non-relativistic velocities by r I = 1.0%?
An electron accelerated by a potential difference V = 1.0 kV moves in a uniform magnetic field at an angle a = 30° to the vector B whose modulus is B = 29 mT. Find the pitch of the helical trajectory of the electron.
A slightly divergent beam of non-relativistic charged particles accelerated by a potential difference V propagates from a point A along the axis of a straight solenoid. The beam is brought into focus at a distance 1 from the point A at two successive values of magnetic induction B1 and B2. Find the
A non-relativistic electron originates at a point A lying on the axis of a straight solenoid and moves with velocity v at an angle a to the axis. The magnetic induction of the field is equal to B. Find the distance r from the axis to the point on the screen into which the electron strikes. The
From the surface of a round wire of radius a carrying a direct current I an electron escapes with a velocity vo perpendicular to the surface. Find what will be the maximum distance of the electron from the axis of the wire before it turns back due to the action of the magnetic field generated by
A non-relativistic charged particle flies through the electric field of a cylindrical capacitor and gets into a uniform transverse magnetic field with induction B (Fig. 3.100). In the capacitor the particle moves along the arc of a circle, in the magnetic field, along a semi-circle of radius r. The
Uniform electric and magnetic fields with strength E and induction B respectively are directed along the g axis (Fig. 3.101). A particle with specific charge q/m leaves the origin O in the direction of the x axis with an initial non-relativistic velocity re. Find:(a) The coordinate yn of the
A narrow beam of identical ions with specific charge q/m, possessing different velocities, enters the region of space, where there are uniform parallel electric and magnetic fields with strength E and induction B, at the point O (see Fig. 3.101). The beam direction coincides with the x axis at the
A non-relativistic proton beam passes without deviation through the region of space where there are uniform transverse mutually perpendicular electric and magnetic fields with E = 120 kV/m and B = 50 mT. Then the beam strikes a grounded target. Find the force with which the beam acts on the target
Non-relativistic protons move rectilinearly in the region of space where there are uniform mutually perpendicular electric and magnetic fields with E = 4.0 kV/m and B = 50 mT. The trajectory of the protons lies in the plane xz (Fig. 3.t02) and forms an angle φ = 30° with the x axis. Find
A beam of non-relativistic charged particles moves without deviation through the region of space A (Fig. 3.103) where there are transverse mutually perpendicular electric and magnetic fields with strength E and induction B. When the magnetic field is switched off, the trace of the beam on the
A particle with specific charge q/m moves in the region of space where there are uniform mutually perpendicular electric and magnetic fields with strength E and induction B (Fig. 3.104). At the moment t the particle was located at the point O and had zero velocity. For the non-relativistic case
A system consists of a long cylindrical anode of radius a and a coaxial cylindrical cathode of radius b (b < a). A filament located along the axis of the system carries a heating current I producing a magnetic field in the surrounding space. Find the least potential difference between the cathode
Magnetron is a device consisting of a filament of radius a and a coaxial cylindrical anode of radius b which are located in a uniform magnetic field parallel to the filament. An accelerating potential difference V is applied between the filament and the anode. Find the value of magnetic induction
A charged particle with specific charge q/m starts moving in the region of space where there are uniform mutually perpendicular electric and magnetic fields. The magnetic field is constant and has an induction B while the strength of the electric field varies with time as E = Em cos wt, where w =
The cyclotron's oscillator frequency is equal to v = 10 MHz. Find the effective accelerating voltage applied across the dees of that cyclotron if the distance between the neighbouring trajectories of protons is not less than ∆r = 1.0 cm, with the trajectory radius being equal to r = 0.5 m.
Protons are accelerated in a cyclotron so that the maximum curvature radius of their trajectory is equal to r = 50 cm. Find: (a) The kinetic energy of the protons when the acceleration is completed if the magnetic induction in the cyclotron is B = 1.0 T; (b) The minimum frequency of the cyclotron's
Singly charged ions He + are accelerated in a cyclotron so that their maximum orbital radius is r = 60 cm. The frequency of a cyclotron's oscillator is equal to v = 10.0 MHz, the effective accelerating voltage across the dees is V = 50 kV. Neglecting the gap between the dees, find: (a) The total
Since the period of revolution of electrons in a uniform magnetic field rapidly increases with the growth of energy, a cyclotron is unsuitable for their acceleration. This drawback is rectified in a microtron (Fig. 3.105) in which a change AT in the period of revolution of an electron is made
The ill effects associated with the variation of the period of revolution of the particle in a cyclotron due to the increase of its energy are eliminated by slow monitoring (modulating) the frequency of accelerating field. According to what law w (t) should this frequency be monitored if the
A particle with specific charge q/m is located inside a round solenoid at a distance r from its axis. With the current switched into the winding, the magnetic induction of the field generated by the solenoid amounts to B. Find the velocity of the particle and the curvature radius of its trajectory,
In a betatron the magnetic flux across an equilibrium orbit of radius r = 25 cm grows during the acceleration time at practically constant rate Ф = 5.0 Wb/s. In the process, the electrons acquire an energy W = 25 MeV. Find the number of revolutions made by the electron during the acceleration
Demonstrate that electrons move in a betatron along a round orbit of constant radius provided the magnetic induction on the orbit is equal to half the mean value of that inside the orbit (the betatron condition).
Using the betatron condition, find the radius of a round orbit of an electron if the magnetic induction is known as a function of distance r from the axis of the field. Examine this problem for the specific case B = Bo – ar2, where Bo and a are positive constants.
Using the betatron condition, demonstrate that the strength of the eddy-current field has the extremum magnitude on an equilibrium orbit.
In a betatron the magnetic induction on an equilibrium orbit with radius r = 20 cm varies during a time interval At = 1.0 ms at practically constant rate from zero to B = 0.40 T. Find the energy acquired by the electron per revolution.
The magnetic induction in a betatron on an equilibrium orbit of radius r varies during the acceleration time at practically constant rate from zero to B. Assuming the initial velocity of the electron to be equal to zero, find: (a) The energy acquired by the electron during the acceleration time;
A small ball is suspended over an infinite horizontal conducting plane by means of an insulating elastic thread of stiffness k. As soon as the ball was charged, it descended by x cm and its separation from the plane became equal to l. Find the charge of the ball.
A point charge q is located at a distance l from the infinite conducting plane. What amount of work has to be performed in order to slowly remove this charge very far from the plane?
Two point charges, q and --q, are separated by a distance l, both being located at a distance l/2 from the infinite conducting plane. Find: (a) The modulus of the vector of the electric force acting on each charge; (b) The magnitude of the electric field strength vector at the midpoint between
A point charge q is located between two mutually perpendicular conducting half-planes. Its distance from each half-plane is equal to l. Find the modulus of the vector of the force acting on the charge.
A point dipole with an electric moment p is located at a distance l from an infinite conducting plane. Find the modulus of the vector of the force acting on the dipole if the vector p is perpendicular to the plane.
A point charge q is located at a distance l from an infinite conducting plane. Determine the surface density of charges induced on the plane as a function of separation r from the base of the perpendicular drawn to the plane from the charge.
A thin infinitely long thread carrying a charge λ, per unit length is oriented parallel to the infinite conducting plane. The distance between the thread and the plane is equal to l. Find: (a) The modulus of the vector of the force acting on a unit length of the thread; (b) The
A very long straight thread is oriented at right angles to an infinite conducting plane; its end is separated from the plane by a distance l. The thread carries a uniform charge of linear density 4. Suppose the point O is the trace of the thread on the plane. Find the surface density of the induced
A thin wire ring of radius R carries a charge q. The ring is oriented parallel to an infinite conducting plane and is separated by a distance l from it. Find: (a) The surface charge density at the point of the plane symmetrical with respect to the ring; (b) The strength and the potential of the
Find the potential R of an uncharged conducting sphere out- side of which a point charge q is located at a distance l from the sphere's centre.
A point charge q is located at a distance r from the centre O of an uncharged conducting spherical layer whose inside and outside radii are equal to R1 and R2 respectively. Find the potential at the point O if r < R1.
A system consists of two concentric conducting spheres, with the inside sphere of radius a carrying a positive charge ql. What charge q2 has to be deposited on the outside sphere of radius b to reduce the potential of the inside sphere to zero? How does the potential φ depend in this case on a
Four large metal plates are located at a small distance d from one another as shown in Fig. 3.8. The extreme plates are inter-connected by means of a conductor while a potential difference Aq0 is applied to internal plates. Find:(a) The values of the electric field strength between neighbouring
Two infinite conducting plates 1 and 2 are separated by a distance 1. A point charge q is located between the plates at a distance x from plate 1. Find the charges induced on each plate.
Find the electric force experienced by a charge reduced to a unit area of an arbitrary conductor if the surface density of the charge equals (r.
A metal ball of radius R = 1.5 cm has a charge q = 10μC. Find the modulus of the vector of the resultant force acting on a charge located on one half of the ball.
When an uncharged conducting ball of radius R is placed in an external uniform electric field, a surface charge density σ = σ0 cos θ is induced on the ball's surface (here a0 is a constant, 0 is a polar angle). Find the magnitude of the resultant electric force acting on an induced
An electric field of strength E = 1.0 kV/cm produces polarization in water equivalent to the correct orientation of only one out of N molecules. Find N. The electric moment of a water molecule equals p = 0.62,10 -29 C ∙ m.
A non-polar molecule is located at the axis of a thin uniformly charged ring of radius R. At what distance x from the ring's centre is the magnitude of the force F acting on the given molecule (a) Equal to zero; (b) Maximum? Draw the approximate plot Fx (x).
A point charge q is located at the centre of a ball made of uniform isotropic dielectric with permittivity e. Find the polarization P as a function of the radius vector r relative to the centre of the system, as well as the charge q' inside a sphere whoso radius is less than the radius of the ball.
A point charge q is located at the centre of a ball made of uniform isotropic dielectric with permittivity e. Find the polarization P as a function of the radius vector r relative to the centre of the system, as well as the charge q' inside a sphere whoso radius is less than the radius of the ball.
Demonstrate that at a dielectric-conductor interface the surface density of the dielectric's bound charge σ' = – σ (ε – 1)/ε, where e is the permittivity, σ is the surface density of the charge on the conductor.
A conductor of arbitrary shape, carrying a charge q, is surrounded with uniform dielectric of permittivity e (Fig. 3.9). Find the total bound charges at the inner and outer surfaces of the dielectric.
A uniform isotropic dielectric is shaped as a spherical layer with radii a and b. Draw the approximate plots of the electric field strength E and the potential φp vs the distance r from the centre of the layer if the dielectric has a certain positive extraneous charge distributed uniformly:
Near the point A (Fig. 3.10) lying on the boundary between glass and vacuum the electric field strength in Vacuum is equal to E0 = 10.0 V/m, the angle between the vector E0 and the normal n of the boundary line being equal to a0 = 30°. Find the field strength E in glass near the point A, the
Near the plane surface of a uniform isotropic dielectric with permittivity e the electric field strength in vacuum is equal to E0, the vector Eo forming an angle 0 with the normal of the dielectric's surface (Fig. 3.11). Assuming the field to be uniform both inside and outside the dielectric,
An infinite plane of uniform dielectric with permittivity e is uniformly charged with extraneous charge of space density p. The thickness of the plate is equal to 2d. Find: (a) The magnitude of the electric field strength and the potentials functions of distance l from the middle point of the plane
Extraneous charges are uniformly distributed with space density p > 0 over a ball of radius R made of uniform isotropic dielectric with permittivity e. Find: (a) The magnitude of the electric field strength as a function of distance r from the centre of the ball; draw the approximate plots E (r)
A round dielectric disc of radius R and thickness d is statically polarized so that it gains the uniform polarization P, with the vector P lying in the plane of the disc. Find the strength E of the electric field at the centre of the disc if d
Under certain conditions the polarization of an infinite uncharged dielectric plate takes the form P = P0 (1 – x2/d2), where P0 is a vector perpendicular to the plate, x is the distance from the middle of the plate, d is its half-thickness. Find the strength E of the electric field inside the
Initially the space between the plates of the capacitor is filled with air, and the field strength in the gap is equal to E 0. Then half the gap is filled with uniform isotropic dielectric with permittivity e as shown in Fig. 3.12. Find the moduli of the vectors E and D in both parts of the gap (1
Solve the foregoing problem for the case when half the gap is filled with the dielectric in the way shown in Fig. 3.13.
Half the space between two concentric electrodes of a spherical capacitor is filled, as shown in Fig. 3.14, with uniform isotropic dielectric with permittivity e. The charge of the capacitor is q. Find the magnitude of the electric field strength between the electrodes as a function of distance r
Two small identical balls carrying the charges of the same sign are suspended from the same point by insulating threads of equal length. When the surrounding space was filled with kerosene the divergence angle between the threads remained constant. What is the density of the material of which the
A uniform electric field of strength E = 100 V/m is generated inside a ball made of uniform isotropic dielectric with permittivity e = 5.00. The radius of the ball is R = 3.0 cm. Find the maximum surface density of the bound charges and the total bound charge of one sign.
A point charge q is located in vacuum at a distance l from the plane surface, of a uniform isotropic dielectric filling up all the half-space. The permittivity of the dielectric equals e. Find: (a) The surface density of the bound charges as a function of distance r from the point charge q;
Making use of the formulation and the solution of the foregoing problem, find the magnitude of the force exerted by the charges bound on the surface of the dielectric on the point charge q.
A point charge q is located on the plane dividing vacuum and infinite uniform isotropic dielectric with permittivity e. Find the moduli of the vectors D and E as well as the potential φ as functions of distance r from the charge q.
A small conducting ball carrying a charge q is located in a uniform isotropic dielectric with permittivity e at a distance l from an infinite boundary plane between the dielectric and vacuum. Find the surface density of the bound charges on the boundary plane as a function of distance r from the
A half-space filled with uniform isotropic dielectric with permittivity e has the conducting boundary plane. Inside the dielectric, at a distance l from this plane, there is a small metal ball possessing a charge q. Find the surface density of the bound charges at the boundary plane as a function
A plate of thickness h made of uniform statically polarized dielectric is placed inside a capacitor whose parallel plates are inter-connected by a conductor. The polarization of the dielectric is equal to P (Fig. 3.15). The separation between the capacitor plates is d. Find the strength and
A long round dielectric cylinder is polarized so that the vector P = (ar, where (z is a positive constant and r is the distance from the axis. Find the space density p' of bound charges as a function of distance r from the axis.
A dielectric ball is polarized uniformly and statically. Its polarization equals P. Taking into account that a ball polarized in this way may be represented as a result of a small shift of all positive charges of the dielectric relative to all negative charges, (a) Find the electric field strength
Utilizing the solution of the foregoing problem, find the electric field strength E o in a spherical cavity in an infinite statically polarized uniform dielectric if the dielectrics polarization is P, and far from the cavity the field strength is E.
A uniform dialectic ball is placed in a uniform electric field of strength E0. Under these conditions the dielectric becomes polarized uniformly. Find the electric field strength E inside the ball and the polarization P of the dielectric whose permittivity equals e. Make use of the result obtained
An infinitely long round dielectric cylinder is polarized uniformly and statically, the polarization P being perpendicular to the axis of the cylinder. Find the electric field strength E inside the dielectric.
A long round cylinder made of uniform dielectric is placed in a uniform electric field of strength E 0. The axis of the cylinder is perpendicular to vector E0. Under these conditions the dielectric becomes polarized uniformly. Making use of the result obtained in the foregoing problem find the
Find the capacitance of an isolated ball-shaped conductor of radius R1 surrounded by an adjacent concentric layer of dielectric with permittivity e and outside radius R 2
Two parallel-plate air capacitors, each of capacitance C, were connected in series to a battery with emf ε. Then one of the capacitors was filled up with uniform dielectric with permittivity e. How many times did the electric field strength in that capacitor decrease? What amount of charge
The space between the plates of a parallel-plate capacitor is filled consecutively with two dielectric layers 1 and 2 having the thicknesses d1 and d2 and the permittivities el and ε2 respectively. The area of each plate is equal to S. Find: (a) The capacitance of the capacitor; (b) The
The gap between the plates of a parallel-plate capacitor is filled with isotropic dielectric whose permittivity e varies linearly from ex to ε2 (ε2 > ε1) in the direction perpendicular to the plates. The area of each plate equals S, the separation between the plates is equal to d.
Find the capacitance of a spherical capacitor whose electrodes have radii R1 and R2 > R1 and which is filled with isotropic dielectric whose permittivity varies as ε = A/R where a is a constant, and r is the distance from the centre of the capacitor.
A cylindrical capacitor is filled with two cylindrical layers of dielectric with permittivities εl and ε2. The inside radii of the layers are equal to R1 and R2 >R1. The maximum permissible values of electric field strength are equal to Elm and Ezra for these dielectrics. At what
There is a double-layer cylindrical capacitor whose parameters are shown in Fig. 3.16. The breakdown field strength values for these dielectrics are equal to E1 and E2 respectively. What is the breakdown voltage of this capacitor if ε1 R1 E1
Two long straight wires with equal cross-sectional radii a are located parallel to each other in air. The distance between their axes equals b. Find the mutual capacitance of the wires per unit length under the condition b >> a.
A long straight wire is located parallel to an infinite conducting plate. The wire cross-sectional radius is equal to a, the distance between the axis of the wire and the plane equals b. Find the mutual capacitance of this system per unit length of the wire under the condition a
Find the capacitance of a system of two identical metal balls of radius a if the distance between their centers is equal to b, with b >> a. The system is located in a uniform dielectric with permittivity e.
Determine the capacitance of a system consisting of a metal ball of radius a and an infinite conducting plane separated from the centre of the ball by the distance l if l >> a.
Find the capacitance of a system of identical capacitors between points A and B shown in(a) Fig. 3.17a;(b) Fig. 3.17b.
Four identical metal plates are located in air at equal distances d from one another. The area of each plate is equal to S. Find the capacitance of the system between points A and B if the plates are interconnected as shown?(a) In Fig. 3.t8a;(b) In Fig. 3.18b.
A capacitor of capacitance C1 = 1.0μF withstands the maximum voltage V1 = 6.0 kV while a capacitor of capacitance C2 = 2.0μF, the maximum voltage V2 = 4.0 kV. What voltage will the system of these two capacitors withstand if they are connected in series?
Find the potential difference between points A and B of the system shown in Fig. 3.19 if the emf is equal to ε = 110 V and the capacitance ratio C2/C1 = η = 2.0.
Find the capacitance of an infinite circuit formed by the repetition of the same link consisting of two identical capacitors, each with capacitance C (Fig. 3.20).
Find the capacitance of an infinite circuit formed by the repetition of the same link consisting of two identical capacitors, each with capacitance C (Fig. 3.20).
In a circuit shown in Fig. 3.22 find the potential difference between the left and right plates of each capacitor.
Find the charge of each capacitor in the circuit shown in Fig. 3.22.
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