Physics for Scientists and Engineers Chapter 23: Electric Potential Copyright © 2004 by W. H. Freeman & Company Paul A. Tipler Gene Mosca Fifth Edition Lecture 4

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Learning Objectives To provide a definition of Electric Potential V To show how E is calculated from V

dU = -mg.dl dW = mg.dl m m h W=mgh

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

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

This work done by an E-field represents a decrease in the electric potential energy ( dU = -dW ) dW = qE.dl dU = -qE.dl

If the E-field varies as particle moves from a point a to b we need to integrate line integral

a b dldl a b dldl Positive charges move from high field (high U) to low field (low U).

ab dldl For a negative charge, q is -ve, U b -U a is positive. The E-field tends to move the charge from b to a. Negative charges move from low field (high U) to high field (low U). When a charge is moved in an E-field, its potential energy is a function of position. This leads to the definition of the electric potential.

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).

Alessandro Volta ( ) The first electric battery

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Electric Potential Difference (23-1) The electric potential difference between two points is: Most commonly used for electric fields [Volt] = [N/C] [N/C]= [V/m] [m]  V b and V a are unique for points b and a

If the charge returns to its original position, by any route, NO WORK IS DONE The Uniqueness of Electric Potential E-field is conservative (electrostatics)

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

Coulomb Potential q can be positive or negative, so is the potential.

The U of a charge q 0 at a distance r from q is:

Potential due to a collection of point charges (23-2) r i is the distance from the i th charge, q i, to the point at which V is being evaluated

E-field lines and Equipotential Surfaces (23-5) An E-field line traces the path that a +ve test charge would follow under the action of electrostatic forces. If released the positive charge will accelerate in the direction of the electric field

E-field lines and Equipotential Surfaces (23-5) Note that lines of force are always perpendicular to the equipotentials Surfaces over which V is constant are called equipotentials

Why do we use potentials?

Calculation of E from the Electric Potential: the Gradient of Potential

In general The negative of the gradient of the electric potential

In Plane Polar coordinates

The Coulomb potential: This is easy because V is a function of r only.

Summary We can characterise an electric field through the electric potential Electric potentials can be easily superposed (numbers not vectors)

Summary

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:

But potentials are equal as connected: 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

Summary

Next Lecture Further properties of Electrostatic Fields Gauss’s Law

Classwork A positive (small) test charge q 0 in the neighbourhood of other charges experiences an electric force F which varies in magnitude and direction at each point. The electric field strength at any point is defined as

Potential – arbitrary zero of potential taken at infinite separation. Suppose an external agent does work W in bringing a small test charge q 0 from infinity to a particular point in an electric field. The electric potential at that point is defined by the equation:

Electric Potential: Numerically equal to the work done in bringing a unit positive charge to the body from the arbitrary point chosen as the zero of potential (often chosen to be infinite separation). Suppose an external agent does work W in bringing a small test charge q from infinity to a particular point in an electric field. The electric potential V at that point is numerically equal to:

Work done by the gravitational field g is equal to the decrease in P.E. Tipler Fig Work done by the E is equal to the decrease in P.E.

Figure 23-3

23-19

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