Electric Field and Equipotential Lines
The electric force between charged particles is similar to the gravitational force.
Electric charge, like mass, is considered an indefinable quantity. However, unlike
mass, electric charge has both positive and negative nature, which means electric
forces can be both attractive and repulsive. Coulomb law gives the magnitude of the
electric force,
(1)
kq1q2
FE =
r2
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 the test charge is determined and the electric field
is defined as

 F
E=
qT
(2)
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 Fig. 1 and 2.
€
+
-
Fig. 2
Fig. 1
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,
 

E = E1 + E 2
where
kq and
kq as shown in Fig. 3.
E1 =
€
€
1
r12
€
E2 =
2
r22
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If this is done for all possible locations of qT, then the test charge will trace out the lines
of force shown in Fig. 3 (The lines of force are the dashed lines). Notice that the lines of
force start on the positive charge ql and terminate on the negative charge q2 .
E1
E
E2
r1
r2
Fig. 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. In Fig. 3, the equipotential lines are the solid lines.
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
(3)
ΔV
E =− 
Δs
The magnitude of the gradient of a scalar function is the change in the function ( Δ V)
€
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divided by the change in distance Δs . 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 Fig. 3.
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.
Fig. 3 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
2.
Galvanometer
3.
Power Supply
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 Fig. 4.
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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.
Fig. 4
4.
Once the seven equipotential lines are drawn, sketch in the electric lines of
force.
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