Electromagnetism I Lecture 4 Electric Potential Learning Objectives To define Electric Potential V To show how E is calculated from V

dldl a b q Charged Particle in an E-field From STMR: work done = force x distance moved in direction of force dW = F.dl F dW = qE.dl

This work done by an E-field represents a decrease in the electric potential energy (dU = -dW) dU = -qE.dl If the E-field varies as particle moves from a point a to b we need to integrate a line integral Definition of Electric Potential: Potential Energy of the system per unit charge It is a property of a point in an E-field. It is a scalar. The unit of potential, joule coulomb -1, is called a volt (V).

Electric Potential Difference The electric potential difference between two points is: [Volt] =[N/C][N/C]= [V/m][m] V b and V a are unique for points b and a E-field is conservative (electrostatics) If the charge returns to its original position, by any route, NO WORK IS DONE

Electric Potential at a distance r from a point charge q q r = V(r) – V(  ) = V(r) – 0 Coulomb Potential

V due to a collection of point charges r i is the distance from the i th charge, q i, to the point at which V is being evaluated The U of a charge q 0 at a distance r from q is:

E-field lines and Equipotential Surfaces An E-field line traces the path that a +ve test charge would follow under the action of electrostatic forces Note that lines of force are always perpendicular to the equipotentials Surfaces over which V is constant are called equipotentials

Consider taking a small step dl = dx i + dy j + dz k The small change in potential is: Now suppose the displacement is parallel to the x-axis, so dy = dz = 0 Calculation of E from the Electric Potential: the Gradient of Potential

Similarly In general The negative of the gradient of the electric potential In Plane Polar coordinates

Application: Field ion microscope - used to image atoms Physicist’s approximation of a needle is: a Radius b Long conducting wire Q Charge q Works by having a high electric field around a point of a needle. How is this high electric field achieved? Electric potential of larger sphere: Electric potential of smaller sphere: Compare electric fields at the surface of each sphere Smaller radius of curvature, the higher the E-field

Practical Importance: High Voltage Power Lines Losses are higher than normal in damp weather. Why? Charged water droplets on wire become elongated to a point because of repulsion. The resulting high E-field leads to ionisation and heating of the air (energy loss) Results in TV and radio interference Classwork Explain briefly the terms electric field strength and electric potential and describe the relation between them. The electric potential at a distance r from a point charge q is given by Calculate the electric field as a function of r.

Summary We can characterize an electric field through the electric potential energy charges acquire in it Electric potentials can be easily superposed (numbers not vectors) V due to a single point charge q at a distance r from the charge is The electric potential difference between two points a and b is given by the line integral