TOPIC 4 Electrostatic Potential

Potential Energy Consider the familiar case of gravitational potential energy. Gravitational force acts downwards To move an object upwards, an applied external force is required, opposite to the gravitational force Gravitational potential energy increases as an object is moved upwards The zero of gravitational potential energy is arbitrary.

Electrostatic Potential Energy Now consider a charge in an electric field Electrostatic force = qE acts to right. An external force Fext = –qE, does work Wext = Fextx = –qEx in moving q x to the left. Note in this case x is negative, so Wext is positive! The change (increase) in potential energy of q is thus U = –qEx .

Electrostatic Potential The potential energy obviously depends on the size of q. Electrostatic potential V is defined as the potential energy of a unit charge. So Note: Unit of potential is Volt = 1 Joule per Coulomb Zero of potential is arbitrary – e.g. value at infinity

In general, since work done is The potential difference between two points is Example 1 Find the potential about a point charge Q. Example 2 Find the potential about a conducting sphere of radius R carrying charge Q.

Example 3 Find the electrostatic potential both outside and inside the insulating sphere carrying a uniform charge density, as discussed in the previous topic.

Electric Field from Potential We have seen that In the case of a radial field, This can be inverted, to give Example: Use the expression for potential to find the field about a point charge

In general we can write as So Or where Alternative units for E: It is more common to use Volts per metre (V m–1) instead of N C–1.

Potential Energy of a System of Charges Consider two charges, Q and q, separated by r. The potential due to Q at q is The potential energy is We can also consider the system as a potential due to q at Q: So the potential of Q is BUT U is the total energy of the system of the two charges!! Do not add two contributions.

Potential Energy of Continuous Charge Distribution How much potential energy is stored in a distribution of charge, e.g. a charged conducting sphere? Consider adding a small charge dq to a sphere already carrying Q. The increase in potential energy is To build up the a charge Qtot from zero, the increase is Note integrating gives a factor ½!! Why doesn’t U = Q V(R)??

Motion of free charges A free charge in an electric field will move due to the electrostatic force. Total energy is conserved – it will lose potential energy U and gain kinetic K. K + U = 0 Since U = q V, K = –q V. Electron-volt – unit of energy 1 eV = change in kinetic energy when an electron (of charge e) moves through potential difference of 1 V. 1 eV = 1.610–19 J

Example 4 A proton is released from rest in a uniform electric field of magnitude 8.0104 V m–1. The proton travels a distance of 0.5 m in the direction of E. (a) Find the change in electric potential between start and end points. (b) Find the change in potential energy of the proton-field system as a result of this displacement. (c) Find the speed of the proton at the end point.

Example 5 Four equal charges, of size Q, are positioned at the corners of a square of side a. What is their potential energy due to electrostatic interactions? (Take U = 0 when the charges are infinitely spaced.)