Physics homework #11

1. If we launch an electron into the uniform electric field

−

→

E with an initial horizontal velocity

−

→

v

0

, what

is the equation of its trajectory, y(x) =? (

−

→

E is vertical, perpendicular to

−

→

v

0

; neglect gravity)

2. Two equal point masses with equal positive charges q = 1µC each are separated by the distance

x = 10 cm. What is the value of these masses such that the net force (gravity force + electric force)
acting on each mass is zero? Assume that there are no external fields (gravitational, electric etc.).

3. Point charges q

1

and q

2

of +12 nC and −12 nC, respectively, are placed 0.10 m apart. Compute the

electric field at the three points, placed: (a) 6 cm from q

1

and 4 cm from q

2

, (b) 4 cm from q

1

and

14 cm from q

2

, (c) 13 cm from q

1

and 13 cm from q

2

.

4. Determine the point on the line joining two charges q

1

and q

2

placed a distance l apart at which the

electric field is zero.

5. A charged cork ball of mass m = 1 g is suspended on a light string in the presence of a uniform

electric field. When

−

→

E = (3ˆ

x + 5ˆ

y) × 10

5

N/C, the ball is in equilibrium and the angle between the

string and the vertical is θ = 37

◦

. Find the charge on the ball and the tension in the string.

6. Show that the potential energy for a dipole in an electric field equals U = −

−

→

p ·

−

→

E .

7. A thin ring-shaped conductor with radius a carries a total charge Q uniformly distributed around

it. (a) Show that the electric field at a point P that lies on the axis of the ring at a distance x from
its centre is equal E =

kQx

(x

2

+a

2

)

3/2

. (b) What is the approximate result if x a?

8. Positive electric charge Q is distributed uniformly along a line with length 2a, lying along the y-axis

between y = −a and y = +a. (a) Find the electric field at a point P on the x-axis at a distance x
from the origin. (b) Find the result if a → +∞, with the charge per unit length equal λ.

9. We place positive charge q an a solid conducting sphere with radius R. Find

−

→

E at any point inside

or outside the sphere. Graph the electric-field magnitude E as a function of r.

10. Positive charge q is distributed uniformly throughout the volume of an insulating sphere with radius

R. Find the magnitude of the electric field at a point P a distance r from the centre of the sphere
(0 < r < +∞). Graph the electric-field magnitude E as a function of r.

11. Electric charge is distributed uniformly along a infinitely long, thin wire. The charge per unit length

is λ (assumed positive). Find the electric field at a distance r from the wire. Graph the electric-field
magnitude E as a function of r.

12. A non-uniform, but spherically symmetric, distribution of charge as a charge density ρ(r) is given

as follows: ρ(r) = ρ

0

(1 − 4r/3R) for r ≤ R, and ρ(r) = 0 for r ≥ R, where ρ

0

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 ≥ R. (c) Obtain an expression for the electric field in the region r ≤ 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.

13. Two point charges are located on the x-axis, q

1

= −e at x = 0 and q

2

= +e at x = a. (a) Find the

work that must be done by an external force to bring a third point charge q

3

= +e from infinity to

x = 2a. (b) Find the total potential energy of the system of three charges.

Maciej Wo loszyn

WFiIS AGH

http://fatcat.ftj.agh.edu.pl/~woloszyn/phys/