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A capacitor \(C\) is initially uncharged. At time \(t=0,\) the capacitor is attached through a resistor \(R\) to a battery. The energy stored in the capacitor increases, eventually reaching a value \(U\) as \(t \rightarrow \infty .\) After a time equal to the time constant \(\tau=R C\), the energy stored in the capacitor is given by a) Ule. c) \(U(1-1 / e)^{2}\) b) \(U / e^{2}\), d) \(U(1-1 / e)\).

Short Answer

Expert verified
In an RC circuit where a resistor R is connected in series with an uncharged capacitor C, the energy stored in the capacitor after a time τ (=RC) is given by the function U(τ) = U(1-1/e)², where U is the final energy stored in the capacitor when it is fully charged. The correct answer is option (c): \(U(1-1 / e)^{2}\).

Step by step solution

01

Find the voltage across the capacitor as a function of time

To find the voltage across the capacitor as a function of time, we will use the formula: \(V(t) = V_0 (1 - e^{-\frac{t}{RC}})\) where \(V(t)\) is the voltage across the capacitor at time \(t\), \(V_0\) is the battery voltage, \(R\) is the resistance, \(C\) is the capacitance, and \(e\) is the base of the natural logarithm.
02

Find the energy stored in the capacitor as a function of time

The energy stored in a capacitor is given by the formula: \(U(t) = \frac{1}{2} CV^2(t)\) Substitute the voltage function from Step 1 into the energy formula: \(U(t) = \frac{1}{2}C[V_0 (1 - e^{-\frac{t}{RC}})]^2\)
03

Calculate the energy stored in the capacitor after time τ

We are given the time constant τ: \(τ = RC\) The energy stored in the capacitor after time τ is: \(U(τ) = \frac{1}{2}C[V_0 (1 - e^{-\frac{τ}{RC}})]^2\) Substitute τ: \(U(τ) = \frac{1}{2}C[V_0 (1 - e^{-\frac{RC}{RC}})]^2\) Simplify and find the energy stored in the capacitor: \(U(τ) = \frac{1}{2}C[V_0 (1 - e^{-1})]^2\)
04

Calculate the final energy stored in the capacitor as t → ∞

As t → ∞, the capacitor will be fully charged, and the energy stored in the capacitor will reach its maximum value U: \(U = \frac{1}{2}C[V_0]^2\)
05

Compare the answer choices to find the correct option

Now we have the expressions for the energy stored in the capacitor after time τ and the final energy stored in the capacitor after infinite time. We need to compare the given answer choices to find the correct one. a) Ule -> This option is not correct since it is not an expression in terms of U. b) \(U / e^{2}\) -> This option is not correct since it is not equal to the derived expression of \(U(τ)\). c) \(U(1-1 / e)^{2}\) -> This is the correct option as it matches the derived expression for \(U(τ)\). d) \(U(1-1 / e)\) -> This option is not correct since it doesn't match the derived expression for \(U(τ)\). So, the correct answer is choice (c) \(U(1-1 / e)^{2}\).

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