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Two parallel non-conducting rings arranged with their central axes along a common line. Ring 1
has uniform charged –q and radius R: ring 2 has uniform charge q and the same radius R. The
rings are separated by a distance 2R.
a) With V= 0 at infinity, derive an expression for the electric potential V at point P on the
common line, at a distance x from the origin O.
b) Derive an expression for the net electric filed E (magnitude and direction) at point P?
Three point charges q1= 5 μC, q2 = 4 μC , and q3 = 1 μC are shown in the figure.
a. With V= 0 at infinity, find the electric potential V at point P located on the x axis at x = 2 m.
b. A proton is placed at point P (x = 2 m), the speed of the proton when it is very far from the three
An electron moving parallel to the x axis has an initial speed of 3.70 × 106 m/s at the origin. Its
speed is reduced to 1.40 × 105 m/s at the point x = 2.00 cm. Calculate,
(a) the potential difference between the origin and that point.
(b) if the electric in the region between the origin and the x = 2.00 cm is constant, what the
magnitude and the direction of this electric field.
(c) Which point is at the higher potential?
A thin ring of radius R carries a positive charge Q spread uniformly over its circumference.
(a) Starting from the definition of the potential of a point charge, derive an expression for the
potential created by the ring at a distance x from its center along the axis of the ring.
(b) If an electron is released from rest at x R 3 , how fast will it be moving when it reaches the
Four point charges, q 2 nC each, are fixed in position at the corners of a square 3 m on a side.
Calculate the electrostatic potential energy (in electron volts) of an electron placed at the center
of the square. nC 10 -9 C, 1 eV 1.6 10 19 J .
A proton (mass = 1.67 10–27 kg, charge = 1.60 10–19 C) moves from point A to point B under the
influence of an electrostatic force only. At point A the proton moves with a speed of 60 km/s. At
point B the speed of the proton is 80 km/s. Determine the potential difference V B V A . [Hint: use
conservation of energy.]
Charge Q is distributed uniformly over a non-conducting ring of radius R .
a) Derive an expression for the potential V (z ) at a point on the axis of the ring a
distance z from its center.
b) Use the expression derived above to find the electric field strength E z at the same
point. k e 9 10 9 Nm2 /C 2 , e 1.6 10 19 C .
Evaluate V and E z for Q due to a million electrons, R 3 cm and z 4 cm.
The figure shows an annulus of inner radius a = 2.0 cm and an outer radius b = 5.0 cm. The
annulus has a uniform charge surface density = 1.5×106C/m2.
a) With V= 0 at infinity calculate the electric potential V at point C the center of
b) What is the magnitude of the electric field at point C?
c) What is the total charge q on the surface?
Two charges q1 = 5 C and q2 = +2C are located at the opposite corners of a rectangle.
With V= 0 at infinity, and A and B are the at opposite corners of the rectangle, as shown,
a) what is the electric potential at point A?
b) What is the electric potential at point B?
c) How much work is required to move a third charge q3 = +3C from point B to
point A along the diagonal of the rectangle?
1. A charged particle (q= – 8.0 mC), which moves in a region where the only force acting
on the particle is an electric force, is released from rest at point A. At point B the kinetic
energy of the particle is equal to 4.8 J. What is the electric potential difference (VBVA)?
2. If a = 30 cm, b = 20 cm, q = + 2.0 nC and Q = – 3.0 nC in the figure, what is the
potential difference (VA-VB)?
A uniform electric field of magnitude 325 V/m is directed in the negative y direction in
the Figure. The coordinates of point A are
(–0.200, –0.300) m, and those of point B
are (0.400, 0.500) m. Calculate the potential difference VB – VA, using the shown path.
Given two 2.00-μC charges, as shown in the Figure, What is the electrical potential at
the origin due to the two 2.00-μC charges?
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