Question

What are the magnitude and direction of the electric field $\overrightarrow{\mathbf{E}}$at location $<20,0,0>\mathbf{cm}$if there is a negative point charge of $\mathbf{1}\mathbf{nC}\left(1\times {10}^{-9}C\right)$at location$<40,0,0>\mathbf{cm}$? Include units.

Short answer

The magnitude of the electric field is and its direction is negative.

Step-by-step solution

Identification of given data

The given data is listed below as:

• The Value of the negative point charge is,$q=1\times {10}^{-9}C$
• The position of the electric field is,
• The position of charge is,

Definition of Electric field for charge particle

The electric field is described as a region which helps a charged particle to exert force on another charged particle.

The electric field inside a sphere having uniform charge is expressed as:

$\mathbf{E}\mathbf{=}\frac{\mathbf{1}}{\mathbf{4}{\mathbf{\pi \epsilon }}_{\mathbf{0}}}\mathbf{-}\frac{\mathbf{q}}{\left|\overrightarrow{r}\right|}\hat{\mathbf{r}}$ …(i)

Here,$\mathbf{q}$is the charge of the sphere,$\frac{\mathbf{1}}{\mathbf{4}{\mathbf{\pi \epsilon }}_{\mathbf{0}}}$is the electric field constant with value $\mathbf{9}\mathbf{\times }{\mathbf{10}}^{\mathbf{9}}\mathbf{N}\mathbf{.}{\mathbf{m}}^{\mathbf{2}}\mathbf{/}\mathbf{C}$,$\left|\overrightarrow{r}\right|$is the magnitude of the position vector and$\hat{\mathbf{r}}$is the unit vector.

Calculation of the unit vector

The equation of the value of the position vector is expressed as:

$\overrightarrow{r}=r_2-Ir$

Here,$r_1$is the location of the charge and $r_2$is the position of the electric field.

Substitute the values in the above equation.

The equation of the magnitude of the position magnitude of vector is expressed as:

$\left|\overrightarrow{r}\right|=\sqrt{{\left(X_1\right)}^2+{\left(y_1\right)}^2+{\left(Z_1\right)}^2}$

Here,$x_1$is the position of the point in the $x$coordinate, $y_1$is the position of the point in the $y$coordinate and role="math" localid="1656919307064" $z_1$is the position of the point in the z coordinate.

Substitute the values in the above equation.

The equation of the unit vector is expressed as:

$\hat{r}=\frac{\overrightarrow{r}}{\left|\overrightarrow{r}\right|}$

Here,$\overrightarrow{r}$is the position vector and $\left|\overrightarrow{r}\right|$is the magnitude of the position vector.

Substitute the values in the above equation.

Determination of the electric field

Substitute all the values in the equation (i).

Thus, the magnitude the electric field is .