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Step-by-Step Solution
Step 1: Identify the Magnetic Field Due to the Larger Coil (C₂)
The magnetic field at the center of a circular coil of radius R_2 carrying current I(t) with N_2 turns is given by:
B_2 = \frac{\mu_0 \, N_2 \, I(t)}{2 \, R_2}.
Here,
- N_2 = 200 (number of turns of coil C₂),
- R_2 = 20\,\text{cm} = 20 \times 10^{-2}\,\text{m} ,
- I(t) = 5t^2 - 2t + 3 (time-dependent current in amperes),
- \mu_0 = 4 \pi \times 10^{-7}\,\text{H/m} .
Step 2: Calculate the Magnetic Flux Through the Smaller Coil (C₁)
Since coil C₁ (with N_1 = 500 turns and radius R_1 = 1\,\text{cm} = 10^{-2}\,\text{m} ) is concentric with C₂, it experiences the same magnetic field B_2 at its center. The flux ( \phi ) through one turn of C₁ is:
\phi_{\text{single turn}} = B_2 \times \text{area of C₁} = B_2 \times \pi R_1^2.
Accounting for N_1 turns in C₁, the total flux becomes:
\phi = N_1 \, B_2 \,\pi R_1^2.
Substituting B_2 from Step 1, we get:
\phi = N_1 \, N_2 \, \frac{\mu_0 I(t)}{2 R_2} \, \pi R_1^2.
Step 3: Substitute the Numerical Values
Substitute N_1 = 500,\; N_2 = 200,\; \mu_0 = 4\pi \times 10^{-7},\; R_2 = 20 \times 10^{-2},\; R_1 = 10^{-2},\; I(t) = 5t^2 - 2t + 3 :
\phi
= \frac{500 \times 200 \times 4 \pi \times 10^{-7} \times (5t^2 - 2t + 3) \times \pi (10^{-2})^2}{2 \times (20 \times 10^{-2})}.
Simplifying the constants, one eventually obtains:
\phi = \bigl(5t^2 - 2t + 3\bigr)\,\times 10^{-4}.
Step 4: Find the Induced EMF (Faraday's Law)
Faraday's law states that the induced EMF e is the negative rate of change of flux. We take the magnitude:
e = \left| \frac{d\phi}{dt} \right|.
Given
\phi = \bigl(5t^2 - 2t + 3\bigr)\,\times 10^{-4},
we differentiate with respect to t :
\frac{d\phi}{dt} = \left( 10t - 2 \right) \times 10^{-4}.
Therefore,
e = \bigl(10t - 2\bigr) \times 10^{-4}.
Step 5: Evaluate the EMF at t = 1\,\text{s}
Substitute t = 1 :
e = \bigl(10 \times 1 - 2\bigr) \times 10^{-4}
= 8 \times 10^{-4}\,\text{V}.
Converting to millivolts,
e = 0.8\,\text{mV}.
Step 6: Relate 0.8 mV to the Given Form 4/x
The problem states that the induced emf at t = 1\,\text{s} is 4/x mV. We have calculated e = 0.8\,\text{mV} . Thus:
\frac{4}{x} = 0.8.
Solving for x :
x = \frac{4}{0.8} = 5.
Final Answer
The required value of x is 5.
