Two different coils have inductances L_1 = 4 | vedclass

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(C) Given: $\frac{dI_1}{dt} = \frac{dI_2}{dt} = k$ (constant rate).Power $P = eI = (L \frac{dI}{dt})I$.Given $P_1 = 4P_2$,so $L_1 \frac{dI_1}{dt} I_1

Vedclass · Accepted Answer

$\frac{U_1}{U_2} = 2 \left( \frac{I_1}{I_2} \right) = \left( \frac{e_1}{e_2} \right)^3$

Vedclass · Answer

(C) Given: $\frac{dI_1}{dt} = \frac{dI_2}{dt} = k$ (constant rate).Power $P = eI = (L \frac{dI}{dt})I$.Given $P_1 = 4P_2$,so $L_1 \frac{dI_1}{dt} I_1 = 4 L_2 \frac{dI_2}{dt} I_2$.Since $\frac{dI_1}{dt} = \frac{dI_2}{dt}$,we have $L_1 I_1 = 4 L_2 I_2$.Substituting $L_1 = 4 \ mH$ and $L_2 = 2 \ mH$: $4 I_1 = 4(2) I_2 \implies I_1 = 2 I_2 \implies \frac{I_1}{I_2} = 2$.Potential difference $e = L \frac{dI}{dt}$,so $\frac{e_1}{e_2} = \frac{L_1}{L_2} = \frac{4}{2} = 2$.Energy stored $U = \frac{1}{2} L I^2$,so $\frac{U_1}{U_2} = \frac{L_1 I_1^2}{L_2 I_2^2} = \left( \frac{L_1}{L_2} \right) \left( \frac{I_1}{I_2} \right)^2 = (2) (2)^2 = 8$.Checking options: For option $C$,$\frac{U_1}{U_2} = 8$,$2 \left( \frac{I_1}{I_2} \right) = 2(2) = 4$,and $\left( \frac{e_1}{e_2} \right)^3 = 2^3 = 8$. Note: The provided option $C$ expression $\frac{U_1}{U_2} = 2(\frac{I_1}{I_2}) = (\frac{e_1}{e_2})^3$ is mathematically consistent with the derived values $8 = 4 = 8$ is incorrect. Re-evaluating: $\frac{U_1}{U_2} = 8$,$\frac{I_1}{I_2} = 2$,$\frac{e_1}{e_2} = 2$. Thus $\frac{U_1}{U_2} = (\frac{e_1}{e_2})^3 = 8$ and $\frac{U_1}{U_2} = 4(\frac{I_1}{I_2}) = 8$. Option $C$ is the closest match.

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