11.2 Problems
473
11.47 Equal charges (Q) are placed at the four corners of a square of side a.
Show that the force on any charge due to the other three charges is given
by 1.914 Q 2 /4πε 0 a 2 .
11.48 (a) Calculate the electric field due to a dipole on its perpendicular bisector.
(b) Show that for the distance x >> d/2, where d is the distance between
the charges, the field varies inversely as the cube of distance.
(c) A molecule has a dipole moment of 6 × 10 −30 cm. Calculate the difference in potential energy when the dipole is placed parallel to the electric
field of 3 × 10 6 V/m and then antiparallel to the field.
11.2.2 Gauss’ Law
11.49 (a) State Gauss’ law of electrostatics in mathematical form.
(b) Use Gauss’ law to show that the electric field magnitude due to an infinite sheet of charge, carrying a surface density σ , is given by
E =
σ
2ε 0
(c) A small sphere of mass 2 mg carries a charge of 5 × 10 −8 C. It hangs
by a silk thread attached to a vertical uniformly charged sheet such that,
under the influence of both gravity and the electric force, it makes an
angle of 10 ◦ with the sheet. Calculate the surface charge density of the
sheet.
11.50 (a) State Gauss’ law in differential and integral form.
(b) Show that the electric field outside a charged sphere is Q/4πε 0 r 2 , where
r is the distance from the centre of the ball.
(c) Show that the electric field inside a uniformly charged solid sphere, with
total charge Q and radius R, is Qr/4πε 0 R 3 .
11.51 In prob. (11.50) the central part of the sphere is hollowed by creating a cavity
of radius
1
2 R concentric with the original sphere. If the charge density of the
hollowed sphere remains unchanged show that the electric field at the surface
is now 7/8 of the original value on the surface.
11.52 In prob. (11.50) show that the electric potential (a) varies as that to simple
harmonic motion for r < R. (b) V (0) =
3
2
V (R) where V (∞) = 0.
11.53 Figure 11.10 shows a non-conducting hollow sphere with inner radius b and
outer radius a. A total charge Q is uniformly distributed in the material b <
r < a. Find the electric field for (a) r < b; (b) b < r < a; (c) r > a.
11.54 A charge Q is uniformly distributed in a long cylinder of radius R and charge
density ρ. Find the electric field for the regions (a) r > R; (b) r < R.
473
11.47 Equal charges (Q) are placed at the four corners of a square of side a.
Show that the force on any charge due to the other three charges is given
by 1.914 Q 2 /4πε 0 a 2 .
11.48 (a) Calculate the electric field due to a dipole on its perpendicular bisector.
(b) Show that for the distance x >> d/2, where d is the distance between
the charges, the field varies inversely as the cube of distance.
(c) A molecule has a dipole moment of 6 × 10 −30 cm. Calculate the difference in potential energy when the dipole is placed parallel to the electric
field of 3 × 10 6 V/m and then antiparallel to the field.
11.2.2 Gauss’ Law
11.49 (a) State Gauss’ law of electrostatics in mathematical form.
(b) Use Gauss’ law to show that the electric field magnitude due to an infinite sheet of charge, carrying a surface density σ , is given by
E =
σ
2ε 0
(c) A small sphere of mass 2 mg carries a charge of 5 × 10 −8 C. It hangs
by a silk thread attached to a vertical uniformly charged sheet such that,
under the influence of both gravity and the electric force, it makes an
angle of 10 ◦ with the sheet. Calculate the surface charge density of the
sheet.
11.50 (a) State Gauss’ law in differential and integral form.
(b) Show that the electric field outside a charged sphere is Q/4πε 0 r 2 , where
r is the distance from the centre of the ball.
(c) Show that the electric field inside a uniformly charged solid sphere, with
total charge Q and radius R, is Qr/4πε 0 R 3 .
11.51 In prob. (11.50) the central part of the sphere is hollowed by creating a cavity
of radius
1
2 R concentric with the original sphere. If the charge density of the
hollowed sphere remains unchanged show that the electric field at the surface
is now 7/8 of the original value on the surface.
11.52 In prob. (11.50) show that the electric potential (a) varies as that to simple
harmonic motion for r < R. (b) V (0) =
3
2
V (R) where V (∞) = 0.
11.53 Figure 11.10 shows a non-conducting hollow sphere with inner radius b and
outer radius a. A total charge Q is uniformly distributed in the material b <
r < a. Find the electric field for (a) r < b; (b) b < r < a; (c) r > a.
11.54 A charge Q is uniformly distributed in a long cylinder of radius R and charge
density ρ. Find the electric field for the regions (a) r > R; (b) r < R.