A point charge of $\mathbf{3}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{9}}\mathbf{\text{\hspace{0.17em}C}}$is located at the origin.

(a) What is the magnitude of the electric field at location $\mathbf{⟨}\mathbf{0.2}\mathbf{,}\mathbf{0}\mathbf{,}\mathbf{0}\mathbf{⟩}\mathbf{\text{\hspace{0.17em}m}}$?

(b) Next, a short, straight, thin copper wire $\mathbf{3}\mathbf{\text{\hspace{0.17em}mm}}$long is placed along the x axis with its center at location $\mathbf{⟨}\mathbf{0.1}\mathbf{,}\mathbf{0}\mathbf{,}\mathbf{0}\mathbf{⟩}\mathbf{\text{\hspace{0.17em}m}}$. What is the approximate change in the magnitude of the electric field at location $\mathbf{⟨}\mathbf{0.2}\mathbf{,}\mathbf{0}\mathbf{,}\mathbf{0}\mathbf{⟩}\mathbf{\text{\hspace{0.17em}m}}$?

(c) Does the magnitude of the electric field at location $\mathbf{⟨}\mathbf{0.2}\mathbf{,}\mathbf{0}\mathbf{,}\mathbf{0}\mathbf{⟩}\mathbf{\text{\hspace{0.17em}m}}$ increase or decrease as a result of placing the copper wire between this location and the point charge?

(d) Does the copper metal block the electric field contributed by the point charge?

The given data can be listed below as:

• The charge of the point charge is $Q=3\times {10}^{-9}\text{\hspace{0.17em}C}$.
• The location of the electric field is, $\mathrm{r}=⟨0.2,0,0⟩\text{\hspace{0.17em}m}$.
• The length of the copper wire is, $\mathrm{s}=3\text{\hspace{0.17em}mm}\times \left(\frac{{\text{10}}^{\text{-3}}\text{\hspace{0.17em}m}}{\text{1\hspace{0.17em}mm}}\right)=3\times {10}^{-3}\text{\hspace{0.17em}m}$.
• The location of the copper wire is, $\mathrm{d}=⟨0.1,0,0⟩\text{\hspace{0.17em}m}$.

The magnitude of the electric field due to a point charge is directly proportional with the charge and inversely proportional to the square of the distance of the charge to the center of the electric field.

The magnitude of the electric field due to a dipole is directly proportional with the charge of the dipole and the distance of separation and inversely proportional to the cube of the distance of the dipole to the center of the electric field.

The equation of the magnitude of the location of the electric field can be expressed as:

$\mathrm{r}=\sqrt{{\mathrm{r}}_{\mathrm{x}}^2+{\mathrm{r}}_{\mathrm{y}}^2+{\mathrm{r}}_{\mathrm{z}}^2}$

Here, $\mathrm{r}$ is the magnitude of the location of the electric field, ${\mathrm{r}}_{\mathrm{x}}$is the location of the electric field at the $\mathrm{x}$axis, $r_y$is the location of the electric field at the $y$ axis and $r_z$ is the location of the electric field at the role="math" localid="1661327540844" $z$axis.

Substitute the values in the above equation.

$\begin{array}{c}\mathrm{r}=\sqrt{{\left(0.2\right)}^2+{\left(0\right)}^2+{\left(0\right)}^2}\text{\hspace{0.17em}m}\\ \text{=0.2\hspace{0.17em}m}\end{array}$

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

$\mathrm{E}=\mathrm{k}\frac{\mathrm{Q}}{{\mathrm{r}}^2}$

Here, $\mathrm{E}$ is the magnitude of the electric field due to a point charge, $k$ is the electric field constant, $Q$ is the point charge and $r$ is the magnitude of the location of the electric field.

Substitute the values in the above equation.

$\begin{array}{c}\mathrm{E}=\left({\text{9×10}}^{\text{9}}{\text{\hspace{0.17em}Nm}}^{\text{2}}{\text{/C}}^{\text{2}}\right)\frac{{\text{3×10}}^{\text{-9}}\text{\hspace{0.17em}C}}{{\left(\text{0.2\hspace{0.17em}m}\right)}^{\text{2}}}\\ =\frac{{\text{18\hspace{0.17em}Nm}}^{\text{2}}\text{/C}}{\left({\text{0.04\hspace{0.17em}m}}^2\right)}\\ =675\text{\hspace{0.17em}N/C}\end{array}$

Thus, the magnitude of the electric field at location $⟨0.2,0,0⟩\text{\hspace{0.17em}m}$ is $675\text{\hspace{0.17em}N/C}$.

As the wire behaves like a conductor, at equilibrium, the net electric field is zero. The equation of the net electric field can be expressed as:

$\begin{array}{c}{\mathrm{E}}_{\text{net}}={\mathrm{E}}_{\text{ext}}+{\mathrm{E}}_{\text{pol}}\\ =0\end{array}$

Here, $E_{\text{net}}$ is the magnitude of the net electric field, $E_{\text{ext}}$is the magnitude of the external electric field and $E_{\text{pol}}$ is the magnitude of the polar electric field.

The above equation can also be expressed as:

$0=E_{\text{ext}}+E_{\text{negative\hspace{0.17em}plate}}+E_{\text{positive\hspace{0.17em}plate}}$ …(i)

Here, $E_{\text{ext}}$ is the magnitude of the external electric field and $E_{\text{negative\hspace{0.17em}plate}}$is the magnitude of the electric field of the negative plate and $E_{\text{positive\hspace{0.17em}plate}}$ is the magnitude of the electric field of the positive plate.

The equation of the magnitude of the external electric field is expressed as:

$E_{\text{ext}}=k\frac{Q}{r^2}$ …(ii)

Here, $E_{\text{ext}}$ is the magnitude of the external electric field due to a point charge, $k$is the electric field constant, $Q$is the point charge and $r$is the magnitude of the location of the electric field.

The equation of the magnitude of the negative plate electric field is expressed as:

${\mathrm{E}}_{\text{negative\hspace{0.17em}plate}}=-\mathrm{k}\frac{{\mathrm{Q}}_1}{{(\mathrm{s}/2)}^2}$ … (iii)

Here, ${\mathrm{E}}_{\text{negative\hspace{0.17em}plate}}$ is the magnitude of the electric field of the negative plate, $k$ is the electric field constant, $Q_1$is the charge of the induced dipole and $s$is the length of the copper wire.

The equation of the magnitude of the positive plate electric field is expressed as:

${\mathrm{E}}_{\text{positive\hspace{0.17em}plate}}=-\mathrm{k}\frac{{\mathrm{Q}}_1}{{(\mathrm{s}/2)}^2}$ …(iv)

Here, ${\mathrm{E}}_{\text{positive\hspace{0.17em}plate}}$ is the magnitude of the electric field of the positive plate, $k$ is the electric field constant, ${\mathrm{Q}}_1$ is the charge of the induced dipole and $s$is the length of the copper wire.

Substitute the values of the equation (ii), (iii) and (iv) in the equation (i).

$\begin{array}{c}0=\frac{\mathrm{Q}}{{\mathrm{r}}^2}-\mathrm{k}\frac{{\mathrm{Q}}_1}{{(\mathrm{s}/2)}^2}-\mathrm{k}\frac{{\mathrm{Q}}_1}{{(\mathrm{s}/2)}^2}\\ \frac{\mathrm{Q}}{{\mathrm{r}}^2}=\frac{2{\mathrm{Q}}_1}{{(\mathrm{s}/2)}^2}\\ {\mathrm{Q}}_1=\frac{\mathrm{Q}}{8}{\left(\frac{\mathrm{s}}{\mathrm{r}}\right)}^2\end{array}$

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

$\begin{array}{c}{\mathrm{E}}_1=\mathrm{k}\frac{{\mathrm{Q}}_1\mathrm{s}}{{\mathrm{r}}^3}\\ =\mathrm{k}\frac{\mathrm{s}}{{\mathrm{r}}^3}\cdot \frac{\mathrm{Q}}{8}{\left(\frac{\mathrm{s}}{\mathrm{r}}\right)}^2\end{array}$

Here, $E_1$ is the change in the magnitude of the electric field, $\mathrm{k}$ is the electric field constant, $\mathrm{r}$is the magnitude of the location of the electric field and $\mathrm{s}$ is the length of the copper wire.

Substitute the values in the above equation.

$\begin{array}{c}{E}_1=(9\times {10}^9\text{\hspace{0.17em}N}\cdot {\text{m}}^{\text{2}}{\text{/C}}^{\text{2}})\frac{\left({\text{3×10}}^{\text{-3}}\text{\hspace{0.17em}m}\right)}{{\left(\text{0.2\hspace{0.17em}m}\right)}^{\text{3}}}\cdot \frac{3\times {10}^{-9}\text{\hspace{0.17em}C}}{8}{\left(\frac{{\text{3×10}}^{\text{-3}}\text{\hspace{0.17em}m}}{0.1\text{\hspace{0.17em}m}}\right)}^2\\ =\frac{({\text{27×10}}^6\text{\hspace{0.17em}N}\cdot {\text{m}}^3{\text{/C}}^2)}{(8\times {10}^{-3}{\text{\hspace{0.17em}m}}^3)}\cdot 3\times {10}^{-9}\text{\hspace{0.17em}C}\left(\frac{{\text{9×10}}^{\text{-6}}{\text{\hspace{0.17em}m}}^2}{0.01{\text{\hspace{0.17em}m}}^2}\right)\\ =\left({\text{3.3×10}}^9{\text{\hspace{0.17em}N/C}}^2\right)\cdot 2.7\times {10}^{-12}\text{\hspace{0.17em}C}\\ \approx 9.1\times {10}^{-3}\text{\hspace{0.17em}N/C}\end{array}$

Thus, the approximate change in the magnitude of the electric field at location $⟨0.2,0,0⟩\text{\hspace{0.17em}m}$ is $9.1\times {10}^{-3}\text{\hspace{0.17em}N/C}$.

The change in the magnitude due to the copper wire at location $⟨0.2,0,0⟩\text{\hspace{0.17em}m}$ is $9.1\times {10}^{-3}\text{\hspace{0.17em}N/C}$. Hence, it has been identified that the magnitude of the electric field increases as it consists that value.

Thus, the magnitude of the electric field at location $⟨0.2,0,0⟩\text{\hspace{0.17em}m}$ increases.

The point charge is mainly helpful for finding the magnitude of the net electric field due to a point charge. The copper metal block has its own electric field and the point charge does not contribute in the electric field.

Thus, the point charge does not contribute to the electric field of the copper metal block.