Question

An infinitely long solenoid, with its axis along $\hat{k}$, carries a current $I$. In addition, there is a uniform line charge density $\lambda$ along the axis. If $\vec{S}$ is the energy flux, in cylindrical coordinates $(\hat{\rho}, \hat{\phi}, \hat{k})$, then
 

• A. $\vec{S}$ is along $\hat{\rho}$
• B. $\vec{S}$ is along $\hat{k}$
• C. $\vec{S}$ has non zero components along $\hat{\rho}$ and $\hat{k}$
• D. $\vec{S}$ is along $\hat{\rho}\times \hat{k}$

Hint:

If $\vec{E}$ is radial and $\vec{B}$ is axial, the Poynting vector must wrap azimuthally around the axis.

Answer 1

The Correct Option is D

Step 1: Recall Poynting vector.
$\vec{S} = \dfrac{1}{\mu_0}\vec{E}\times \vec{B}.$

Step 2: Identify directions of $\vec{E$ and $\vec{B}$.}
• A line charge along the axis produces an electric field radially outward: $\vec{E}\propto \hat{\rho}$.
• A solenoid produces a magnetic field along the axis: $\vec{B}\propto \hat{k}$.

Step 3: Take the cross product.
$\vec{E}\times \vec{B} \propto \hat{\rho}\times \hat{k}$.
In cylindrical coordinates, this is the azimuthal direction $\hat{\phi}$.

Step 4: Conclusion.
Energy flows in the $\hat{\phi}$ (circulating) direction around the axis ⇒ option (D).