Question

Consider a solenoid of length \( L \) and radius \( R \), where \( R \ll L \). A steady current flows through the solenoid. The magnetic field is uniform inside the solenoid and zero outside. 
Among the given options, choose the one that best represents the variation in the magnitude of the vector potential, \( (0, A_\phi, 0) \) at \( z = L/2 \), as a function of the radial distance \( r \) in cylindrical coordinates. 

Useful information: The curl of a vector \( \mathbf{F} \), in cylindrical coordinates is \[ \nabla \times \mathbf{F}(r, \phi, z) = \hat{r} \left[ \frac{1}{r} \frac{\partial F_z}{\partial \phi} - \frac{\partial F_\phi}{\partial z} \right] + \hat{\phi} \left[ \frac{\partial F_r}{\partial z} - \frac{\partial F_z}{\partial r} \right] + \hat{z} \left[ \frac{1}{r} \left( \frac{\partial}{\partial r} (r F_\phi) \right) - \frac{1}{r} \frac{\partial F_r}{\partial \phi} \right] \]

• A. A
• B. B
• C. C
• D. D

Hint:

In a solenoid, the magnetic vector potential increases linearly with the radial distance up to the solenoid radius \( r = R \).

Answer 1

The Correct Option is C

For a solenoid, the vector potential \( \mathbf{A} \) is related to the magnetic field \( \mathbf{B} \) by the relation \( \mathbf{B} = \nabla \times \mathbf{A} \). For an ideal solenoid, the magnetic field inside is uniform and can be expressed as: \[ \mathbf{B} = \hat{z} B_z = \hat{z} \frac{\mu_0 N I}{L} \] where \( \mu_0 \) is the permeability of free space, \( N \) is the number of turns, \( I \) is the current, and \( L \) is the length of the solenoid. Step 1: Magnetic vector potential in cylindrical coordinates The magnetic vector potential \( A_\phi \) for a solenoid is given by the solution to Ampere's law. Inside the solenoid, the vector potential is azimuthal (\( A_\phi \)) and depends on the radial distance \( r \): \[ A_\phi(r) = \frac{\mu_0 N I r}{2 L} \] This equation indicates that \( A_\phi \) increases linearly with \( r \) inside the solenoid, and reaches a maximum at \( r = R \), the radius of the solenoid. Step 2: Behavior of the vector potential At the fixed point \( z = L/2 \), the vector potential is strongest at the outer edge of the solenoid (\( r = R \)) and weaker at the center (\( r = 0 \)). Therefore, the correct graph that represents this variation is one where \( A_\phi \) increases with \( r \) and reaches a maximum at \( r = R \). This matches option (C). Thus, the correct answer is (C).