If the magnetic vector potential \(A=-\frac{\rho^2}{4} a_z\) Wb/m, what is the total magnetic flux crossing the surface \(\phi=\frac{\pi}{2}\), 1 ≤ p ≤ 2m, 0 ≤ z ≤ 5m?

Magnetic vector potential A = \(-\frac{\rho^2}{4}a_z\)

\(1\leq\rho\leq2, 0 \leq Z\leq5\)

\(\vec B = \vec\nabla \times\vec A\)

\(\vec B =\frac{1}{\rho}[-\rho a\phi(\frac{-2\rho}{4})]\)

\(\vec B = \frac{\rho}{2}a\phi\)

\(\psi = \int\int \vec B \cdot \vec ds = \int\int \frac{\rho}{2}a_\phi \cdot d\rho dZa\phi\)

\(\psi = \int^2_1\int^5_0 \frac{\rho}{2} d\rho dZ\)

\(\psi = [\frac{\rho^2}{4}]^2_1 [Z]^5_0\)

\(\psi = [\frac{4-1}{4}]\times 5 = \frac{15}{4} = 3.75 Wb\)

Here, option 3 is correct.