Physics tutorial #14
1. A non-uniform, but spherically symmetric, distribution of charge as a charge density Á(r) is given
as follows: Á(r) = Á0(1 - 4r/3R) for r d" R, and Á(r) = 0 for r e" R, where Á0 is a positive constant.
(a) Find the total charge contained in the charge distribution. (b) Obtain an expression for the
electric field in the region r e" R. (c) Obtain an expression for the electric field in the region r d" R.
(d) Graph the electric-field magnitude E as a function of r. (e) Find the value of r at which the
electric field is maximum, and find the value of that maximum field.
2. (a) An insulting sphere with radius a has a uniform charge density Á. The sphere is not centred at
the the origin (x = 0, y = 0) but at r = b. Show that the electric field inside the sphere is given by
Á
E = ( r - b)3 .
0
(b) An insulating sphere of radius R has a spherical hole of radius a located within its volume and
centred a distance b from the centre of the sphere, where a < b < R. The solid part of the sphere has
a uniform volume charge density Á. Find the magnitude and direction of the electric field E inside
the hole, and show that E is uniform over the entire hole. [Hint: Use the principle of superposition
and the result of part (a).]
3. Two point charges are located on the x-axis, q1 = -e at x = 0 and q2 = +e at x = a. (a) Find the
work that must be done by an external force to bring a third point charge q3 = +e from infinity to
x = 2a. (b) Find the total potential energy of the system of three charges.
4. Find the potential at a distance r from a very long line of charge with linear charge density (charge
per unit length) . Is V (") = 0 a good choice in this case?
q
5. The potential at a radial distance r from a point charge is V = . Find the vector electric field
4Ä„ r
0
from this expression for V .
6. A proton is released from rest in a uniform electric field that has a magnitude of 8.0 × 104 V/m.
The proton undergoes a displacement of 0.5 m in the direction of E. (a) Find the change in electric
potential between the staring point and the final point. (b) Find the change in potential energy of
the proton field system for this displacement. (c) Find the speed of the proton after completing the
0.5 m displacement in the electric field.
7. A small sphere with mass 1.50 g hangs by a thread between two parallel vertical plates 5.0 cm apart.
The plates are insulating and have uniform surface charge densities +Ã and -Ã. The charge on the
sphere is q = 8.9 × 10-6 C. What potential difference between the plates will cause the thread to
assume an angle of 30ć% with the vertical?
8. The electric potential V in a region of space is given by V (x, y, z) = A(x2 - 3y2 + x2) where A is
a constant. (a) Derive an expression for the electric field E at any point in this region. (b) The
work done by the field when a 1.50-µC test charge moves from the point (x, y, z) = (0, 0, 0.250 m)
to the origin is measured to be 6.0 × 10-5 J. Determine A. (c) Determine the electric field at the
point (0, 0, 0.250 m). (d) Show that in every plane parallel to the xz-plane the equipotential contours
are circles. (e) What is the radius of the equipotential contour corresponding to V = 1280 V and
y = 2.0 m ?
Nivas Babu Selvaraj, Maciej Woloszyn
http://fatcat.ftj.agh.edu.pl/~woloszyn/phys/