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.

검증된 단계별 안내

Start by recalling the relationship between charge (Q), capacitance (C), and potential difference (V) for a capacitor:

Q = C \(\cdot\) 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.

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 \(\times\) 10^{-19} \ \(\text{C}\)

, the total charge is given by

Q = N \(\cdot\) e

Substitute the expression for Q from the second step into the capacitor equation

Q = C \(\cdot\) V

. This gives:

N \(\cdot\) e = C \(\cdot\) V

Rearrange the equation to solve for the potential difference (V):

V = \(\frac{N \cdot e}{C}\)

. This shows that the potential difference is proportional to the number of electrons (N), with the proportionality constant being

\(\frac{e}{C}\)

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 = \(\frac{N \cdot e}{C}\)

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.