A charged capacitor is discharged through a resistor - Edexcel - A-Level Physics - Question 3 - 2023 - Paper 1

To determine the time it takes for the potential difference across a capacitor to halve, we use the formula for the discharge of a capacitor given by: $V(t) = V_0 e^{-t/T}$ When the potential difference is halved, we have: $\frac{V_0}{2} = V_0 e^{-t/T}$ Dividing both sides by $V_0$ gives: $\frac{1}{2} = e^{-t/T}$ Taking the natural logarithm of both sides: $\ln(\frac{1}{2}) = -\frac{t}{T}$ Rearranging yields: $t = -T \ln(\frac{1}{2})$ Since $\ln(\frac{1}{2}) = -\ln 2$, substituting gives: $t = T \ln 2$ Thus, the correct answer is D) $t = T \ln 2$.