Question

If the charge of the point charge in Figure 13.60 were $\mathbf{-}\mathbf{9}\mathbf{Q}$(instead of $\mathbf{Q}$):

(a) By what factor would the magnitude of the force on the point charge due to the dipole change? Express your answer as the ratio (magnitude of new force / magnitude of ${\mathbf{F}}_{\mathbf{V}}$).

(b) Would the direction of the force change?

Short answer

a. The magnitude of the force on that point will change by a factor of 9.

It is given as$\left|\frac{{\overrightarrow{\mathrm{F}}}_{\mathrm{new}}}{{\mathrm{F}}_{\mathrm{v}}}\right|=9$

b. Yes, the direction of forces is opposite on the new charge.

Step-by-step solution

Formula used to solve the question.

• The electric field at a location on the dipole axis:

$\mathbf{\left|}{\overrightarrow{\mathbf{E}}}_{\mathbf{axis}}\mathbf{\right|}\mathbf{=}\frac{\mathbf{1}}{\mathbf{4}{\mathbf{\pi \epsilon }}_{\mathbf{0}}}\mathbf{.}\frac{\mathbf{2}\mathbf{qs}}{{\mathbf{r}}^{\mathbf{3}}}$

Where q is the one charge of the dipole, s is the distance between two charges of dipole, r is the distance of the point where we need to find the electric field and$\mathbf{r}\mathbf{>}\mathbf{>}\mathbf{s}$.

• Ifrole="math" localid="1661317742015" $\overrightarrow{\mathbf{E}}$is electric field at some location, then the electric force$\overrightarrow{\mathbf{F}}$acting on any charge Qwe place there is given as

$\overrightarrow{\mathbf{F}}\mathbf{=}\mathbf{Q}\overrightarrow{\mathbf{E}}$

Finding the magnitude of force change.

It is given that $\mathrm{d}\gg \mathrm{s}$.

Therefore, we can use the formula of the electric field produced by dipole given as,

$\left|{\overrightarrow{\mathrm{E}}}_{\mathrm{axis}}\right|=\frac{1}{4{\mathrm{\pi \epsilon }}_0}.\frac{2\mathrm{qs}}{{\mathrm{d}}^3}$

We know that,

If $\overrightarrow{\mathrm{E}}$ is electric field at some location, then the electric force $\overrightarrow{\mathrm{F}}$acting on any charge Qwe place there is given as,

$\overrightarrow{\mathrm{F}}=\mathrm{Q}\overrightarrow{\mathrm{E}}$

The forces acting on the charge $+\mathrm{Q}$ can be given as,

role="math" localid="1661318584471" $\begin{array}{c}\overrightarrow{{\mathrm{F}}_{\mathrm{v}}}=\mathrm{Q}\overrightarrow{\mathrm{E}}\\ \overrightarrow{{\mathrm{F}}_{\mathrm{v}}}=\mathrm{Q}\left[\frac{1}{4{\mathrm{\pi \epsilon }}_0}.\frac{2\mathrm{qs}}{{\mathrm{d}}^3}\right]\\ {\overrightarrow{\mathrm{F}}}_{\mathrm{v}}=\frac{1}{4{\mathrm{\pi \epsilon }}_0}.\frac{2\mathrm{qsQ}}{{\mathrm{d}}^3}\end{array}$

Now, the forces acting on the charge $-9\mathrm{Q}$ can be given as,

$\begin{array}{l}{\overrightarrow{\mathrm{F}}}_{\mathrm{new}}=-9\mathrm{Q}\overrightarrow{\mathrm{E}}\\ {\overrightarrow{\mathrm{F}}}_{\mathrm{new}}=-9\mathrm{Q}\left[\frac{1}{4{\mathrm{\pi \epsilon }}_0}.\frac{2\mathrm{qs}}{{\mathrm{d}}^3}\right]\\ {\overrightarrow{\mathrm{F}}}_{\mathrm{new}}=-\frac{1}{4{\mathrm{\pi \epsilon }}_0}.\frac{2\mathrm{qs}\left(9\mathrm{Q}\right)}{{\mathrm{d}}^3}\end{array}$

Therefore,

$\begin{array}{l}\frac{{\overrightarrow{\mathrm{F}}}_{\mathrm{new}}}{{\mathrm{F}}_{\mathrm{v}}}=\frac{\frac{1}{4{\mathrm{\pi \epsilon }}_0}.\frac{2\mathrm{qsQ}}{{\mathrm{d}}^3}}{-\frac{1}{4{\mathrm{\pi \epsilon }}_0}.\frac{2\mathrm{qs}\left(9\mathrm{Q}\right)}{{\mathrm{d}}^3}}\\ \frac{{\overrightarrow{\mathrm{F}}}_{\mathrm{new}}}{{\mathrm{F}}_{\mathrm{v}}}=-9\end{array}$

This implies that, the magnitude of force at that point will change by a factor of 9.

$\left|\frac{{\overrightarrow{\mathrm{F}}}_{\mathrm{new}}}{{\mathrm{F}}_{\mathrm{v}}}\right|=9$

Finding the direction of new force.

The force on the charge $+\mathrm{Q}$is

${\overrightarrow{\mathrm{F}}}_{\mathrm{v}}=\frac{1}{4{\mathrm{\pi \epsilon }}_0}.\frac{2\mathrm{qsQ}}{{\mathrm{d}}^3}$

The force on charge $-9\mathrm{Q}$ is

${\overrightarrow{\mathrm{F}}}_{\mathrm{new}}=-\frac{1}{4{\mathrm{\pi \epsilon }}_0}.\frac{2\mathrm{qs}\left(9\mathrm{Q}\right)}{{\mathrm{d}}^3}$

From above calculation we can conclude that the direction of forces is apposite on the new charge.