If the vector potential $ \vec{A} = a x \hat{x} + 2y \hat{y} - 3z \hat{z}$, satisfies the Coulomb gauge, the value of the constant $a$ is ___________
The problem asks for the value of the constant $a$ given a vector potential $ \vec{A} = ax \hat{x} + 2y \hat{y} - 3z \hat{z} $ that satisfies the Coulomb gauge condition.
Vector Potential Divergence Calculation
The Coulomb gauge condition is defined as the divergence of the vector potential being zero:
$ \nabla \cdot \vec{A} = 0 $
The divergence of a vector field $ \vec{A} = A_x \hat{x} + A_y \hat{y} + A_z \hat{z} $ in Cartesian coordinates is given by:
$ \nabla \cdot \vec{A} = \frac{\partial A_x}{\partial x} + \frac{\partial A_y}{\partial y} + \frac{\partial A_z}{\partial z} $
For the given vector potential, the components are:
- $ A_x = ax $
- $ A_y = 2y $
- $ A_z = -3z $
Now, calculate the partial derivatives:
- $ \frac{\partial A_x}{\partial x} = \frac{\partial (ax)}{\partial x} = a $
- $ \frac{\partial A_y}{\partial y} = \frac{\partial (2y)}{\partial y} = 2 $
- $ \frac{\partial A_z}{\partial z} = \frac{\partial (-3z)}{\partial z} = -3 $
Substitute these into the divergence formula:
$ \nabla \cdot \vec{A} = a + 2 + (-3) = a - 1 $
Applying Coulomb Gauge Condition
To satisfy the Coulomb gauge, we set the divergence equal to zero:
$ a - 1 = 0 $
Solving for Constant '$a$'
Solving the equation for $a$ gives:
$ a = 1 $
This value $a=1$ falls within the specified range of 0.99 to 1.01.
Important Questions from Magnetostatics Ampere's Law Vector Potential
- Consider a particle of mass $m = 9.0 \times 10^{-5} \text{ kg}$ and charge $q = 3.0 \times 10^{-4} \text{ C}$ in a uniform electromagnetic field $\vec{E} = 2\hat{x} \text{ V.m}^{-1}$, $\vec{B} = 3\hat{z} \text{ V.m}^{-2}\text{.s}$. The particle is released from the coordinates $(0, 5\text{ m}, 0)$ at time $t = 0$. Starting from initial speed zero, it comes back to the $y$-axis for the first time at time $t$. The value of $t$ in seconds (rounded off to two decimal places) is _____
The wire loop shown in the figure carries a steady current $I$. Each straight section of the loop has length $d$. A part of the loop lies in xy plane and the other part is tilted at $30^\circ$ with respect to the xz plane. The magnitude of the magnetic dipole moment of the loop (in appropriate units) is
- A linear magnetic material in the form of a cylinder of radius $R$ and length $L$ is placed with its axis parallel to the $z$ axis. The cylinder has uniform magnetization $\vec{M} \hat{k}$. Which of the following option(s) is/are correct?
- Consider an isolated magnetized sphere of radius $R$ with a uniform magnetization $\vec{M}$ along the positive $z$ direction, with the north and south poles of the sphere lying on the $z$ axis. It is given that the magnetic field inside the sphere is $\vec{B} = \frac{2\mu_0}{3} \vec{M}$, where $\mu_0$ is the permeability of vacuum. Which of the following statements is(are) CORRECT?
An infinitely long cylinder of radius $R$ carries a frozen-in magnetization $\vec M = ke^{-s^2} \hat z$, where $k$ is a constant and $s$ is the distance from the axis of cylinder. The magnetic permeability of free space is $\mu_o$. There is no free current present anywhere. The magnetic flux density ($\vec B$) inside the cylinder is
