Capacitors: parallel plate, cylindrical, spherical. Today’s agenda: Capacitance. You must be able to apply the equation C=Q/V. Capacitors: parallel plate, cylindrical, spherical. You must be able to calculate the capacitance of capacitors having these geometries, and you must be able to use the equation C=Q/V to calculate parameters of capacitors. Circuits containing capacitors in series and parallel. You must understand the differences between, and be able to calculate the “equivalent capacitance” of, capacitors connected in series and parallel.

Parallel Plate Capacitance -Q +Q We previously calculated the electric field between two parallel charged plates: E V0 V1 This is valid when the separation is small compared with the plate dimensions. d A We also showed that E and V are related: This lets us calculate C for a parallel plate capacitor.

Reminders: Q is the magnitude of the charge on either plate. V is actually the magnitude of the potential difference between the plates. V is really |V|. Your book calls it Vab. C is always positive.

Parallel plate capacitance depends “only” on geometry. -Q +Q This expression is approximate, and must be modified if the plates are small, or separated by a medium other than a vacuum (lecture 9). V0 d V1 A Greek letter Kappa. For today’s lecture (and for exam 1), use Kappa=1. Do not use =9x109! Because it isn’t!

Coaxial Cylinder Capacitance We can also calculate the capacitance of a cylindrical capacitor (made of coaxial cylinders).  L The next slide shows a cross-section view of the cylinders.

Lowercase c is capacitance per unit length: Gaussian surface b r a Q E -Q dl This derivation is sometimes needed for homework problems! (Hint: 24.10, 11, 12.) Some necessary details are not shown on this slide! See lectures 4 and 6. Lowercase c is capacitance per unit length:

This has not been one of those “useless physics problems that my professor tries to confuse me with.”  L If you are purchasing or specifying coaxial cables, capacitance per length is a critical part of the specifications. example

Isolated Sphere Capacitance An isolated sphere can be thought of as concentric spheres with the outer sphere at an infinite distance and zero potential. We already know the potential outside a conducting sphere: The potential at the surface of a charged sphere of radius R is so the capacitance at the surface of an isolated sphere is

Capacitance of Concentric Spheres If you have to calculate the capacitance of a concentric spherical capacitor of charge Q… In between the spheres (Gauss’ Law) b a +Q -Q You need to do this derivation if you have a problem on spherical capacitors! (not this semester) If there is spherical capacitor homework, details will be provided in lecture!

Example: calculate the capacitance of a capacitor whose plates are 20 cm x 3 cm and are separated by a 1.0 mm air gap. d = 0.001 area = 0.2 x 0.03 If you keep everything in SI (mks) units, the result is “automatically” in SI units.

*Remember, it’s the potential difference that matters. Example: what is the charge on each plate if the capacitor is connected to a 12 volt* battery? 0 V V= 12V +12 V *Remember, it’s the potential difference that matters. If you keep everything in SI (mks) units, the result is “automatically” in SI units.

Example: what is the electric field between the plates? 0 V V= 12V E d = 0.001 +12 V If you keep everything in SI (mks) units, the result is “automatically” in SI units.

Demo: Professor Tries to Avoid Spot-Welding His Fingers to the Terminals of a Capacitor While Demonstrating Energy Storage Asynchronous lecture students: we’ll try to make a video of this.