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A large circular ring (ring 1) of radius R carries a steady current I. A much smaller ring (ring 2) of radius r (r << R) moves with constant velocity V along the axis of ring 1, with its plane always parallel to the plane of ring 1. Find the maximum EMF induced in ring 2.

1. (24 / (25*sqrt(5))) * (mu0 * pi * I * r² * V / R²)
2. (24 / 25) * (mu0 * pi * I * r² * V / R²)
3. (2 / (3*sqrt(3))) * (mu0 * pi * I * r² * V / R²)
4. (48 / (25*sqrt(5))) * (mu0 * pi * I * r² * V / R²)

Correct answer: (24 / (25*sqrt(5))) * (mu0 * pi * I * r² * V / R²)

Solution

Flux through ring 2: phi = B(x)*pi*r² = [mu0*I*R² / (2*(R²+x²)^(3/2))] * pi*r². EMF = d(phi)/dt = (d(phi)/dx)*V. So EMF = pi*r²*V * d/dx[mu0*I*R² / (2*(R²+x²)^(3/2))]. dB/dx = mu0*I*R²/2 * (-3/2)*2x*(R²+x²)^(-5/2) = -3*mu0*I*R²*x / (2*(R²+x²)^(5/2)). |dB/dx| is maximised when d/dx[x/(R²+x²)^(5/2)] = 0. Let f = x*(R²+x²)^(-5/2). f' = (R²+x²)^(-5/2) + x*(-5/2)*2x*(R²+x²)^(-7/2) = 0. (R²+x²) - 5x² = 0. R² = 4x². x = R/2. At x = R/2: R²+x² = R²+R²/4 = 5R²/4. (R²+x²)^(5/2) = (5R²/4)^(5/2) = (5/4)^(5/2)*R⁵ = 5^(5/2)/4^(5/2)*R⁵ = 25*sqrt(5)/(32)*R⁵. |dB/dx|_max = 3*mu0*I*R²*(R/2) / (2*25*sqrt(5)*R⁵/32) = 3*mu0*I*R³/2 * 32/(2*25*sqrt(5)*R⁵) = 3*mu0*I*16/(25*sqrt(5)*R²) = 48*mu0*I/(25*sqrt(5)*R²). EMF_max = pi*r²*V*48*mu0*I/(25*sqrt(5)*R²) = 48*mu0*pi*I*r²*V/(25*sqrt(5)*R²). Hmm that's option D. Let me recheck: 3*mu0*I*R²*(R/2) / (2*(5R²/4)^(5/2)). (5R²/4)^(5/2) = (5/4)^(5/2)*R⁵. (5/4)^(5/2) = 5^(5/2)/4^(5/2) = 25sqrt(5)/32. So denominator = 2*25sqrt(5)/32*R⁵ = 25sqrt(5)/16 * R⁵. Numerator = 3*mu0*I*R³/2. |dB/dx| = (3/2)*mu0*I*R³ / (25sqrt(5)/16 * R⁵) = (3/2)*(16/(25sqrt(5)))*mu0*I/R² = 24*mu0*I/(25*sqrt(5)*R²). EMF_max = pi*r²*V * 24*mu0*I/(25*sqrt(5)*R²) = 24*mu0*pi*I*r²*V/(25*sqrt(5)*R²). This matches option A.

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