Q55P - Introduction to Electrodynamics

Question

Suppose an electric field $\mathit{E}\left(x,y,z\right)$ has the form

                              ${\mathit{E}}_{\mathbf{x}}\mathbf{=}\mathit{a}\mathit{x}\mathbf{,}\mathbf{ }\mathbf{ }\mathbf{ }{\mathit{E}}_{\mathbf{y}}\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }{\mathit{E}}_{\mathbf{z}}\mathbf{=}\mathbf{0}$ 

Where $\mathit{a}$ is a constant. What is the charge density? How do you account for the fact that the field points in a particular direction, when the charge density is uniform?

Step-by-Step Solution

Answer

The formula of charge density is$\rho ={\epsilon }_0\left[\nabla .E\right]$ .

Here, $E$is linear function of x,y and z. From this it is clear that electric field points in particular direction.  

Step 1: Define functions

Write the expression for electric filed from divergence theorem

 

$\mathbf{\nabla }\mathbf{.}\mathit{E}\mathbf{=}\frac{\mathbf{\rho }}{{\mathbf{\epsilon }}_{\mathbf{0}}}$                                    …… (1)

Here,$E$ is the electric filed, $\rho$ is the electric filed,${\epsilon }_0$ is the permittivity for the free space.

Step 2: Determine charge density

Write the formula for electric filed component along with x-axis.

 

$E=ax\stackrel{⏜}{x}$                                    …… (2)

 

Write the expression for charge density.

 

$\rho ={\epsilon }_0\left[\nabla .E\right]$                                …… (3)

 

$\nabla .E=\frac{\partial E_x}{\partial x}+\frac{\partial E_y}{\partial x}+\frac{\partial E_z}{\partial x}$         …… (4)

 

The electric field component along y and z axis is zero. Therefore,$\nabla .E$ is expressed as,

 

$\nabla .E=\frac{\partial E_x}{\partial x}$                              …… (5)

Substitute the value$\frac{\partial E_x}{\partial x} \mathrm{for} \nabla .E$ in equation (3)

 $\rho ={\epsilon }_0=\left[\frac{\partial E_x}{\partial x}\right]$

 

Differentiate $E_x$and consider the value from equation (2),

$$\begin{gathered} \frac{\partial E_x}{\partial x}=\frac{\partial }{\partial x}\left(ax\right) \\ =a \end{gathered}$$ 

Substitute for$\frac{\partial E_x}{\partial x}$ in equation (3)

 $\rho ={\epsilon }_{0}a$

From the above equation, it is clear that $\rho$ is constant everywhere.

Thus, the charge density is $\rho =\mathrm{Constant}$ .

Step 3: Determine charge density is uniform

Write the expression for charge density by equation (3)

$\rho ={\epsilon }_0\nabla .E$ 

            

From the above equation the charge density is directly proportional to the electric filed.

 If charge density is uniform then,         

 $\nabla .E=Cons\tan t$

Therefore, The values of $\frac{\partial E_x}{\partial x}+\frac{\partial E_y}{\partial x}+\frac{\partial E_z}{\partial x}$ are also constant.

 

Since $E$ is linear function of  $x,y \mathrm{and} \mathrm{z}$. From this it is clear that electric field points in particular direction.