Reading Question 27.21 When an initially uncharged capacitor is charged in an RC
ID: 3280273 • Letter: R
Question
Reading Question 27.21 When an initially uncharged capacitor is charged in an RC circuit, what happens to the potential differences across the resistor and capacitor? It increases across the resistor and is constant across the capacitor. It increases across the resistor and decreases across the capacitor. It is constant across the resistor and decreases across the capacitor. It decreases across the resistor and increases across the capacitor. It is constant across the resistor and increases across the capacitor. It decreases across the resistor and is constant across the capacitor.Explanation / Answer
consider a series RC circuit with a battery, resistor, and capacitor in series. The capacitor is initially uncharged, but starts to charge when the switch is closed. Initially the potential difference across the resistor is the battery emf, but that steadily drops as the potential difference across the capacitor increases.
Applying Kirchoff's loop rule:
- IR - Q/C = 0
As Q increases I decreases, but Q changes because there is a current I. As the current decreases Q changes more slowly.
I = dQ/dt, so the equation can be written:
- R (dQ/dt) - Q/C = 0
This is a differential equation that can be solved for Q as a function of time. The solution (derived in the text) is:
Q(t) = Qo [ 1 - e-t/ ]
where Qo = C and the time constant = RC.
Differentiating this expression to get the current as a function of time gives:
I(t) = (Qo/RC) e-t/ = Io e-t/
where Io = /R is the maximum current possible in the circuit.
The time constant = RC determines how quickly the capacitor charges. If RC is small the capacitor charges quickly; if RC is large the capacitor charges more slowly.
so it decreases across the resistor and increases across the capacitor.
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