Electric Field and Equipotential Lines
The electric field due to any charge distribution can be found by placing a test charge
qT at a point in space. The force on this test charge is determined and the electric
field is defined as
E = F / qT
(1)
where E is the electric field and F is the force. The test charge is defined to be
positive so that the direction of the electric field is away from positive charges and
towards negative charges as shown in Figures 1 and 2.
Figure 1
Figure 2
For two equal charges of opposite signs (a dipole), the electric field is the vector sum
of the electric fields due to each charge. Therefore, (EQUATION 1) where
(EQUATION 2)
and (EQUATION 3) as shown in Figure 3.
If this is done for all possible locations of qT, then the test charge will trace out the
lines of force shown in Figure 3. Notice that the lines of force start on the positive
charge ql and terminate on the negative charge q2 .
Figure 3
The change in electric potential is defined as the work done in moving a charge in an
electric field divided by magnitude of the charge. The electric potential is a scalar
quantity while the electric field is a vector quantity. An equipotential line is a line
where the electric potential is constant. Experimentally, equipotential lines can be
found by using a galvanometer. A zero deflection on the galvanometer implies that
the inputs to the galvanometer are at the same potential.
The electric field is the vector gradient of the electric potential, that is
(2)
The magnitude of the gradient of a scalar function is the change in the function ( Δ V)
divided by the change in distance (ΔS). The direction of the gradient is in the
direction of maximum change of ΔV. One property of the gradient operation is that
the lines of force of the electric field are perpendicular to the equipotential line. Also,
the lines of force are perpendicular to the surface of a conductor. This is shown for a
dipole in Figure 4.
Where the equipotential lines are closer together, the electric field is larger. Notice
that the lines of force are l) perpendicular to the surface of the conductor 2)
perpendicular to the equipotential line and 3) start on a positive charge and terminate
on a negative charge.
Figure 4 may be explained by a gravitational analogy. Think of the positive charge as
a hill and the negative charge as a valley. The equipotential lines are topographical
lines of equal elevation around the hill and valley. The lines of force give the
direction of the gravitational force of an object resting on the surface or the direction
a ball would roll if released on the surface.
Equipment
1.
Electric Field apparatus
3.
Power Supply
2. Multimeter
Procedure
1.
There are five field plates that can be used with the electric field apparatus.
They are l) two point charges 2) parallel plate capacitor 3) a point and a plane 4)
insulator and conductor and 5) the Faraday “Ice pail”. You are to map the
equipotential line for the two point charges, the parallel plate capacitor, and a
third field plate of your choice. Mount the board with the blackened side facing
the outward on the electric field apparatus. Be sure to use washers so that the
bolts make good contact with the metallic pattern.
2.
Since there are eight identical resistors mounted across the top of the board, the
voltage between the terminals is divided into 7 equal parts. Connect the
positive terminal of the galvanometer to the junction of one of the resistors and
the negative terminal to the probe as shown in Figure 5.
3.
Attach a sheet of paper to the top of the board. Using a template, which
matches the pattern on the board, trace the pattern onto a piece of paper. Next,
use the probe to find the points on the board that give a zero deflection and
mark these points on the paper. Connecting these points then give one
equipotential line.
When looking for equipotential points, avoid the edges of the board. Repeat for
the seven resistor junctions on the board to get seven equipotential lines.
4.
Once the seven equipotential lines are drawn, sketch in the electric lines of
force.
Questions
1. How much work is done in moving a charge along an equipotential line? Why?
2.
Using the information in this laboratory, sketch the lines of force (including
their directions) for the following charge distributions. All the objects are
conductors.
(a) Two positive point charges:
+
+
(b) An electric dipole (equal positive and negative point charges) with a conducting
plate placed between them. [Hint - what happens inside the conductor?]
+
–
(c) A positive point charge inside a conducting spherical shell.
+
3.
A dipole consists of a positive charge (+q) at (x,y) = (0, a) and a negative charge
(-q) at (0, -a). Find an equation for the electric field at any point (x, 0) along the
x axis. Be sure to indicate the direction of the electric fields.
y
(0,a)
+q
(x,0)
(0,–a)
–q
x