Capacitors can be used as “electric charge counters.” Consider an initially uncharged capacitor of capacitance C with its bottom plate grounded and its top plate connected to a source of electrons. If N electrons flow onto the capacitor’s top plate, show that the resulting potential difference V across the capacitor is directly proportional to N.

Verified step by step guidance

1. Start by recalling the relationship between charge (Q), capacitance (C), and potential difference (V) for a capacitor: $Q\; =\; C\; \backslash (\backslash cdot\backslash )\; V$. This equation tells us that the charge stored on a capacitor is proportional to the product of its capacitance and the potential difference across it.
2. Next, recognize that the charge (Q) on the capacitor is related to the number of electrons (N) that flow onto the top plate. Since each electron has a charge of $e\; =\; 1.6\; \backslash (\backslash times\backslash )\; 10^\{-19\}\; \backslash \; \backslash (\backslash text\{C\}\backslash )$, the total charge is given by $Q\; =\; N\; \backslash (\backslash cdot\backslash )\; e$.
3. Substitute the expression for Q from the second step into the capacitor equation $Q\; =\; C\; \backslash (\backslash cdot\backslash )\; V$. This gives: $N\; \backslash (\backslash cdot\backslash )\; e\; =\; C\; \backslash (\backslash cdot\backslash )\; V$.
4. Rearrange the equation to solve for the potential difference (V): $V\; =\; \backslash (\backslash frac\{N\; \backslash cdot\; e\}\{C\}\backslash )$. This shows that the potential difference is proportional to the number of electrons (N), with the proportionality constant being $\backslash (\backslash frac\{e\}\{C\}\backslash )$.
5. Conclude that the potential difference (V) across the capacitor is directly proportional to the number of electrons (N) that flow onto its top plate, as derived in the equation $V\; =\; \backslash (\backslash frac\{N\; \backslash cdot\; e\}\{C\}\backslash )$.

Key Concepts

Capacitance

Capacitance is a measure of a capacitor's ability to store electric charge per unit voltage. It is defined as the ratio of the charge (Q) stored on one plate of the capacitor to the potential difference (V) across the plates, expressed as C = Q/V. In this context, a capacitor with capacitance C will store charge when electrons are added, affecting the voltage across its plates.

Electric Charge

Electric charge is a fundamental property of matter that causes it to experience a force when placed in an electromagnetic field. In this scenario, electrons, which carry a negative charge, flow onto the capacitor's top plate. The total charge (Q) on the capacitor increases with the number of electrons (N) added, leading to a change in the potential difference across the capacitor.

Potential Difference

Potential difference, or voltage (V), is the work done per unit charge to move a charge between two points in an electric field. For a capacitor, the potential difference is directly related to the amount of charge stored. As more electrons flow onto the capacitor's top plate, the potential difference increases proportionally, illustrating the relationship V = Q/C, where Q is the total charge and C is the capacitance.