Question

A student claimed that the equation for the electric field outside a cube of edge length L, carrying a uniformly distributed charge Q, at a distance x from the center of the cube, was

$\frac{\mathbf{1}}{\mathbf{4}\mathbf{\pi }{\mathbf{\delta }}_{\mathbf{o}}}\frac{\mathbf{50}\mathbf{Q}\mathbf{L}}{{\mathbf{x}}^{\mathbf{3}}}$

Explain how you know that this cannot be the right equation.

Short answer

Answer

The equation should produce the electric field for a point charge for a large distance of the cube from the center of the electric field, but it cannot produce the equation.

Step-by-step solution

Identification of given data

The given data is listed below as:

• The edge length of the cube is, $\mathit{L}$

• The charge of the cube is, $\mathit{Q}$

• The distance of the cube from the center is, $\mathit{x}$

Significance of the magnitude of the electric field

The electric field helps an electrically charged particle to exert force on another particle. The magnitude of the electric field is inversely proportional to the distance of the charged object from the electric field and directly proportional with the charge of that object.

Determination of the correctness of the equation

The equation of the magnitude of the electric field given in the question is expressed as:

$\mathit{E}\mathbf{=}\frac{\mathbf{1}}{\mathbf{4}\mathbf{\pi }{\mathbf{\epsilon }}_{\mathbf{0}}}\frac{\mathbf{50}\mathbf{Q}\mathbf{L}}{{\mathbf{x}}^{\mathbf{3}}}$

Here, $\frac{\mathbf{1}}{\mathbf{4}{\mathbf{\pi \epsilon }}_{\mathbf{0}}}$is the electric field constant, $\mathit{Q}$is the charge of the cube, $\mathit{L}$is the length of the cube and $\mathit{x}$is the distance of the cube from the center of the electric field.

The equation of the magnitude of the electric field for a point charge is expressed as:

${\mathit{E}}_{\mathbf{1}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{4}{\mathbf{\pi \epsilon }}_{\mathbf{0}}}\frac{\mathbf{q}}{{\mathbf{r}}^{\mathbf{2}}}$

Here, $\frac{\mathbf{1}}{\mathbf{4}{\mathbf{\pi \epsilon }}_{\mathbf{0}}}$is the electric field constant, $\mathit{q}$is the charge of an object and $\mathit{r}$is the distance of the object from the center of the electric field.

The expression given in the question is wrong as at a certain distance from the cube, the electric field is approximately same as the electric field having a point charge, but the field is not same.

Thus, the equation should produce the electric field for a point charge for a large distance of the cube from the center of the electric field, but it cannot produce the equation.