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(C) The dynamic resistance $r_d$ is defined as $r_d = \frac{dV}{dI}$.Given $I = I_0 \left( e^{\frac{V}{2 V_T}} - 1 \right)$.For $I \gg I_0$, we can ap

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$1/100$

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(C) The dynamic resistance $r_d$ is defined as $r_d = \frac{dV}{dI}$.Given $I = I_0 \left( e^{\frac{V}{2 V_T}} - 1 \right)$.For $I \gg I_0$, we can approximate $I \approx I_0 e^{\frac{V}{2 V_T}}$.Taking the natural logarithm on both sides: $\ln(I/I_0) = \frac{V}{2 V_T}$, which implies $V = 2 V_T \ln(I/I_0)$.Differentiating with respect to $I$: $\frac{dV}{dI} = 2 V_T \cdot \frac{1}{I/I_0} \cdot \frac{1}{I_0} = \frac{2 V_T}{I}$.Thus, $r_d(I) = \frac{2 V_T}{I}$.We need to find $\alpha$ such that $r_d(1000 I_0) = \alpha r_d(10 I_0)$.Substituting the expression for $r_d$:$\frac{2 V_T}{1000 I_0} = \alpha \cdot \frac{2 V_T}{10 I_0}$.$\frac{1}{1000} = \alpha \cdot \frac{1}{10}$.$\alpha = \frac{10}{1000} = \frac{1}{100}$.

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