# Electric Potential

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This is a topic from Higher Physics 1B

## Introduction

Electric potential is a scalar quantity reflective of an object’s ability to do work and is defined mathematically as follows:

V = U/qo

Where V represents electric potential in Volts (V)

U represents electric potential energy in Joules (J)
qo represents a unit charge (C)

In other words, electric potential is the electric potential energy per unit charge:

Electric Potential = Electric Potential Energy / Charge

The electric potential energy in a system is analogous to the gravitational potential energy in a system, covered in PHYS1121/1131. For the purpose of explanation, if there were to be a concept called 'gravitational potential' it would be defined as the gravitational potential energy per unit mass:

Gravitational Potential = Gravitational Potential Energy / Mass

Consider now two charges, qA and qB:

• The electric potential at qA with respect to itself is zero
• The same can be said for qB
• What is of interest however is the electric potential at qB with respect to qA, or the electric potential difference between the two charges, which is not zero

This electric potential difference is known commonly as Voltage and is given by the following:

Where ds represents a tiny displacement of a charge

The term 'electric potential' is often used interchangeably with the terms 'electric potential difference' and 'voltage', despite the subtle difference between the concepts, and so care must be taken in understanding which concept is meant in the context of the sentence.

## Derivation

Electrical potential difference is most easily considered in terms of the forces on a charge within an electric field of influence, rather than in terms of the forces between individual particles.

• Recalling that when charge qA is placed in an electric field (that of charge qB) it undergoes a force (F=EqA) and that this force is conservative, the work done moving the charge a tiny distance (ds) towards qB is given by
W = Fr
= F ds
= EqA ds
Where W represents work done in Joules (J)
F represents force in Newtons (N)
• Doing work converts potential energy (U) into another form of energy, so
ΔU = -W
= - EqA ds
• When enough of these tiny distances are summed to span the distance A to B,
• Therefore

## Work and Electric Potential

If a charge is moved through an electric field at a constant velocity there is work done on the charge which changes the magnitude of the electric potential energy such that

W = ΔU = qΔV

and thus it takes one Joule of energy to move a charge of one Coulomb through an electric potential difference of one Volt.

## The Electron-Volt

The electron-volt (eV) is a unit of energy equal to the energy gained or lost when an object of charge e (such as a proton or electron) is moved through a potential difference of one volt

1 eV = 1.60 x 10-19J

## Sign Conventions (+/-)

Consider the following example:

Case 1:

• If a positive charge is placed in an electric field, positive work needs to be done on the charge to move it against the direction of this electric field, to the point S1
• This increases the amount of energy/work that would be required to move the charge from the point of reference P to its new position
• Thus the voltage is increased

Case 2:

• If a positive charge is placed in an electric field, negative work needs to be done on the charge for it to move in the direction of this electric field, to the point S2 - (if negative work is being done on the charge, the charge itself is performing positive work)
• This decreases the amount of energy/work that would be required to move the charge from the point of reference P to its new position
• Thus the voltage is decreased

The sign of the potential difference itself is generally considered to be positive.

## Calculations with Electric Potential

The following requires an understanding of the concepts of Electric Potential, Potential Energy and Charge Density.

### Obtaining the Electric Potential between Charged Plates

This is the most straight-forward of the derivations because the electric field between two oppositely charged plates is constant.

Therefore V = E(rB - rA)

V = Ed

### Obtaining the Electric Potential from Point Charges

The electric potential at a given distance from a point charge is given by

V = keq / r

while the electric potential resultant from several charges is given by applying the superposition principle as follows:

V = ke Σi qi / ri

Remembering the definition that V = U/q, for a single point charge the potential energy is given by

U = keq1q2 / r12

while the electric potential for numerous point charges equals the sum of the potential energy between each pair of charges. Therefore system of three point charges, q1, q2 and q3 has the following potential energy:

U = keq1q2 / r12 + keq1q3 / r13 + keq2q3 / r23

Derivations at 25.3 of Serway and Jewett

### Obtaining the Electric Potential from Uniformly Charged Objects

The electric potential at a point P near a charged object such as a ring or a disc is calculated by breaking an object up into an infinite amount of point charges (dq). By summing the resultant voltages of each charge (dq) at their respective distances (r) from P we get the net voltage at P, given by the following integral:

Note the similar method to the derivation of electric fields from uniformly charged objects.

A table of common results is provided below:

Although you are not expected to memorise these results, you are expected to know how to derive them - Derivations at 25.5 of Serway and Jewett

### Obtaining the Electric Field from the Electric Potential

• As seen in the above derivation,
ΔU = - Eqo ds
• And so, using the definition of voltage,
ΔV = - E • ds
dV = - E • ds
• And in cases where the electric field has only one component, Ex, the dot product of E and ds can be rewritten as E dx, and rearranged to give
Ex = - dV / dx
• Thus, by finding an expression for electric potential (V) in terms of distance (x) from source charge, one can differentiate to find an expression for electric field (E)

## Equipotent Surfaces

An equipotent surface is a continuous set of points which all have the same electrical potential, often conveyed as contour lines as in the following figure:
The surface of a charged conductor in electrostatic equilibrium is equipotent, and furthermore, because the electric field is zero inside a charged conductor, the electric potential inside is equal to the electric potential at the surface, as seen in the following figure:

The key when drawing or interpreting equipotent contours is to remember that the contour is perpendicular to the electric field lines passing through it at every point. This is seen in the following figure: