Question

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

role="math" localid="1668495301957" $\mathit{E}\mathbf{=}\frac{\mathbf{Q}}{{\mathbf{\epsilon }}_{\mathbf{0}}}\frac{\mathbf{L}}{{\mathbf{x}}^{\mathbf{1}\mathbf{/}\mathbf{2}}}$

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

Short answer

The claimed equation is incorrect because it gives incorrect dimension of electric field.

Step-by-step solution

Identification of given data

The formula for 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 is claimed by a student to be

$E_c=\frac{Q}{{\epsilon }_0}\frac{L}{x^{1/2}}$

Dimensions of electric field and permittivity

The dimension of electric field is

$\left[E\right]\mathbf{=}{\mathit{M}}^{\mathbf{1}}{\mathit{L}}^{\mathbf{1}}{\mathit{T}}^{\mathbf{-}\mathbf{3}}{\mathit{I}}^{\mathbf{-}\mathbf{1}}$ …(i)

The dimension of permittivity is

$\left[{\epsilon }_0\right]\mathbf{=}{\mathit{M}}^{\mathbf{-}\mathbf{1}}{\mathit{L}}^{\mathbf{-}\mathbf{3}}{\mathit{T}}^{\mathbf{4}}{\mathit{I}}^{\mathbf{2}}$ …(ii)

Determination of the correctness of the claimed formula

The dimension of the claimed formula, using equation (ii), is

$$\begin{gathered} \left[E_c\right]={\left[Q\right]}^1{\left[{\epsilon }_0\right]}^{-1}{\left[L\right]}^1{\left[x\right]}^{-1/2} \\ =I^1{T}^1\times {\left(M^{-1}{L}^{-3}{T}^4{I}^2\right)}^{-1}\times L^1\times L^{-1/2} \\ =M{L}^{7/2}{T}^{-3}{I}^{-1} \end{gathered}$$

This does not match. Hence the formula is incorrect due to incorrect dimension.