Answer :
Given Data: 8 H 9 H 18 H L₁ = 2 H None of these LM = 0.5 H L₂ = 7 H
According to the problem, we have a circuit with two coils and a mutual inductance of LM = 0.5 H, as shown below:
we have the following equations for mutual inductance:
V1 = L₁ di1/dt + M di2/dt
V2 = M di1/dt + L₂ di2/dt
We can rearrange the above two equations as shown below:
di1/dt = [ V1 - M di2/dt ] / L₁
di2/dt = [ V2 - M di1/dt ] / L₂
Differentiating both the above equations with respect to time, we get:
d²i₁/dt² = [-M / L₁] d²i₂/dt²
d²i₂/dt² = [-M / L₂] d²i₁/dt²
Let, the total inductance of the circuit be LT. Then, we can write the equation as follows:
LT d²i₁/dt² = V1 - M di2/dt + LM d²i₂/dt²
LT d²i₂/dt² = V2 - M di1/dt + LM d²i₁/dt²
Now, let's add the above two equations to eliminate d²i/dt² terms:
LT [d²i₁/dt² + d²i₂/dt²] = V1 + V2
We can see that d²i₁/dt² + d²i₂/dt² is the second derivative of the total current with respect to time, i.e., d²i/dt². Therefore, the total inductance of the circuit is given by:
LT = (V1 + V2) / d²i/dt²
We know that for an inductor, the inductance is given by:
L = V / d i/dt
Therefore, we can write the above equation in terms of inductances as follows:
LT = (L₁ + L₂ + 2M + 2LM) / d²i/dt²
Substituting the given values, we get:
LT = (2H + 7H + 2 x 0.5H + 2 x 0.5H) / d²i/dt²
LT = 12 H
Therefore, the total inductance of the circuits is 12 H.
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