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Chapter 15: Electric Potential
9. (Incomplete)

Capacitance

capacitance
\(C\)
Farads
If the potential at infinity is zero, then positively charged conductors have positive potential, and will become more positive as the charge increases. The rate at which the potential increases depends on the shape and size of the conductor. The ratio between the charge on the conductor and its potential is known as the conductor's capacitance $C$:
$$C={Q\over \Delta V}$$

where $Q$ is the charge on the conductor and $\Delta V$ is the difference in potential between the conductor and infinity (where $V=0$). Capacitance is measured in farads; one farad is a coulomb per volt. The larger the capacitance of a conductor, the more charge it can hold for a given potential. Capacitance is useful because potential difference is easier to measure (with a voltmeter) and create (with a battery or power supply) than charge. We usually "charge" a conductor by creating a potential difference $V$ between the conductor and a point at $V=0$, and the charge on the conductor is then $Q=C\Delta V$. Typically, larger conductors have a larger capacitance, because there is more room for the charge to spread out.

For example, a metal sphere with radius $R$ has a capacitance of $$C_{\rm sphere} = \frac1k R$$ where $k=9\ten9\u{F/m}$ is the same constant we saw in Potential Energy of Point Charges. That means that if we charge two spheres to the same potential difference, the larger sphere will have the larger capacitance and end up with more charge. In the example to the right, a 10cm sphere has a capacitance of $C={10\u{cm}\over 9\ten9\u{F/m}} = 11\u{pF}$, and thus has a charge of $Q=C\Delta V = (11\u{pF})(9\u{V}) = 99\u{pC}$. The larger sphere, with $R=20\u{cm}$, has double the capacitance ($C=22\u{pF}$) and ends up with double the charge ($Q=198\u{pC}$).