Outline

Show that the electric field strength can be calculated from the pd.

Class Objectives Show and apply an equation for calculating the electric field strength from the pd.

Calculating the field from the Potential Previously we calculated the potential at a point given the electric field along a given path. Now we wish to do the opposite. That is, to find the electric field given the potential.

Calculating the field from the Potential Consider a test charge q 0 that moves through a displacement ds from one equipotential to another. + E ds s

Calculating the field from the Potential It can be shown that,

Calculating the field from the Potential It can be shown that, In general, In the simple case where the electric field is constant,

Example The electric potential at any point from the central axis of a uniformly charged disc is From this equation derive the expression for the electric field at any point from the central axis.

Example

Capacitance

Outline Introduce capacitors as an application where the method of electric field strength being calculated from the pd will be used.

Capacitance A capacitor is a device which stores energy as potential energy in an electric field.

Capacitance A capacitor is a device which stores energy as potential energy in an electric field. They are used in the VCR, TV, radio etc.

Capacitance A basic capacitor consists of two charge conductors of any shape.

Capacitance When charged the conductors have opposite charge +q and –q.

Capacitance When charged the conductors have opposite charge +q and –q. A common arrangement is the parallel plate capacitor.

Capacitance The charge on a plate is given by,

Capacitance The charge on a plate is given by, Clearly, SI unit is the farad. 1 farad = 1 F = 1 coulomb per volt = 1 C/V

Calculating the Capacitance We now look at calculating the capacitance for a given geometry. Assuming a charge q on the plates, we calculate E. And then V. Given q and V we can determine C.

Capacitance Calculating E. To calculate E we use gauss’ law.

Capacitance We draw a Gaussian surface enclosing the charge on the positive plate.

Capacitance Recall gauss’ law,

Capacitance Recall gauss’ law, Where q is the charge enclosed.

Capacitance Recall gauss’ law, Where q is the charge enclosed. Since E is constant and E and dA are parallel:

Capacitance Calculating V. Recall,

Capacitance Calculating V. Recall, The integral is always in the direction of the electric field. Integrating we get,

Capacitance The two equations produced hold for any geometry.

Capacitance The two equations produced hold for any geometry. Let us continue for the case of a parallel plate capacitor.

Capacitance The two equations produced hold for any geometry. Let us continue for the case of a parallel plate capacitor. Recall:

Capacitance The two equations produced hold for any geometry. Let us continue for the case of a parallel plate capacitor. Recall: Integrating the first equation we get,

Capacitance Substituting for q and V we get that,

Cylindrical Capacitor

Capacitance A cross section of the cylindrical capacitor of length L formed by coaxial cylinders of radii a and b is shown below. a b +q -q

Capacitance Assume L >> b so that fringe effects are negligible. Each plate contains charge q.

Capacitance First determine E. To find E we draw a cylindrical Gaussian surface between the two plates. a b +q -q r

Capacitance Using Gauss’ law we get that, Alt

Capacitance Determining V. Again

Capacitance Determining V. Again Substitute for E,

Capacitance Determining V. Again Substitute for E,

Capacitance Therefore,

Spherical Capacitor

Capacitance Now consider the case of a capacitor made up of two concentric spherical shells of radii a and b. a b +q -q

Capacitance In this case to find E we draw a spherical Gaussian surface between the two shells. a b +q -q r

Capacitance Using Gauss’ law we get that, Alt

Capacitance Determining V. After substituting for E we get,

Capacitance Determining V. After substituting for E we get,

Capacitance Determining V. After substituting for E we get, Which we make more tidy to give,

Capacitance Finally using the equation for the capacitance we get,

Exercise Review capacitors in combination.