Chapter 23, Problem 23.69AP

Three charged particles are aligned along the x axis as shown in Figure P22.35. Find the electric field at (a) the position (2.00 m, 0) and (b) the position (0, 2.00 m).

Figure P22.35

To determine

The electric field at the position $\left(2.00\text{ m, 0}\right)$.

Answer to Problem 23.69AP

The electric field at the position $\left(2.00\text{ m, 0}\right)$ is $24.2\widehat{\text{i}}\text{ N}/\text{C}$

Explanation of Solution

Three charges are acting on the same line along the $x$-axis.

According to Coulomb’s law, write the expression for the electric field created by a charge $q$

    $\overrightarrow{E}=k_e\frac{q}{r^2}\widehat{r}$

Here, $E$ is the electric field, $k_e$ is the Coulomb’s constant, $q$ is the charge and $r$ is the distance between two charges.

The electric field at a point due to number of charges is resultant of the electric field due to the individual charges.

The figure for the position of point charges is shown below.

Figure (1)

Write the expression for the electric field due to charge $q_1$ at $P$

    $\overrightarrow{E_1}=k_e\frac{q_1}{r_1{}^2}{\widehat{r}}_1$

Here, $E_1$ is the electric field due to charge $q_1$ and $r_1$ is the distance between the charge $q_1$ and point $P$.

Substitute $9\times {10}^9{\text{Nm}}^2/{\text{C}}^2$ for $k_e$, $-4.00\text{ nC}$ for $q_1$ and $2.50\text{ m}$ for $r_1$ in above expression.

    $\begin{array}{c}\overrightarrow{E_1}=\frac{\left(9\times {10}^9{\text{Nm}}^2/{\text{C}}^2\right)\left(-4.00\text{ nC}\right)\left(\frac{1\text{C}}{{10}^9\text{ nC}}\right)}{{\left(2.50\text{ m}\right)}^2}\stackrel{\wedge }{i}\\ =-5.76\widehat{\text{i}}\text{N}/\text{C}\end{array}$

Write the expression for the electric field due to charge $q_2$ at $P$

    $\overrightarrow{E_2}=k_e\frac{q_2}{r_2{}^2}{\widehat{r}}_2$

Here, $E_2$ is the electric field due to charge $q_2$ and $r_2$ is the distance between the charge $q_2$ and point $P$.

Substitute $9\times {10}^9{\text{Nm}}^2/{\text{C}}^2$ for $k_e$, $5.00\text{ nC}$ for $q_2$ and $2.00\text{ m}$ for $r_2$ in above expression.

    $\begin{array}{c}\overrightarrow{E_2}=\frac{\left(9\times {10}^9{\text{Nm}}^2/{\text{C}}^2\right)\left(5.00\text{ nC}\right)\left(\frac{1\text{C}}{{10}^9\text{ nC}}\right)}{{\left(2.00\text{ m}\right)}^2}\stackrel{\wedge }{i}\\ =11.25\widehat{\text{i}}\text{N}/\text{C}\end{array}$

Write the expression for the electric field due to charge $q_3$ at $P$

    $\overrightarrow{E_3}=k_e\frac{q_3}{r_3{}^2}{\widehat{r}}_3$

Here, $E_3$ is the electric field due to charge $q_3$ and $r_3$ is the distance between the charge $q_3$ and point $P$.

Substitute $9\times {10}^9{\text{Nm}}^2/{\text{C}}^2$ for $k_e$, $3.00\text{ nC}$ for $q_3$ and $1.20\text{ m}$ for $r_3$ in above expression.

    $\begin{array}{c}\overrightarrow{E_3}=\frac{\left(9\times {10}^9{\text{Nm}}^2/{\text{C}}^2\right)\left(3.00\text{ nC}\right)\left(\frac{1\text{C}}{{10}^9\text{ nC}}\right)}{{\left(1.20\text{ m}\right)}^2}\widehat{\text{i}}\\ =18.75\widehat{\text{i}}\text{N}/\text{C}\end{array}$

Write the expression for the resultant electric field

    $\overrightarrow{E}=\overrightarrow{E_1}+\overrightarrow{E_2}+\overrightarrow{E_3}$

Here, $E$ is the resultant electric field.

Conclusion:

Substitute $-5.76\widehat{\text{i}}\text{N}/\text{C}$ for $\overrightarrow{E_1}$, $11.25\widehat{\text{i}}\text{N}/\text{C}$ for $\overrightarrow{E_2}$ and $18.75\widehat{\text{i}}\text{N}/\text{C}$ for $\overrightarrow{E_3}$.

    $\begin{array}{c}\overrightarrow{E}=\left(-5.76+11.25+18.75\right)\widehat{\text{i}}\text{ N}/\text{C}\\ =24.24\widehat{\text{i}}\text{ N}/\text{C}\end{array}$

Thus, the electric field at position $\left(2.00\text{ m, 0}\right)$ is $24.2\widehat{\text{i}}\text{ N}/\text{C}$.

To determine

The electric field at the position $\left(0,2.00\text{ m}\right)$.

Answer to Problem 23.69AP

The electric field at the position $\left(0,2.00\text{ m}\right)$ is $\left(-4.21\right)\widehat{\text{i}}+\left(8.43\right)\widehat{\text{j}}\text{ N}/\text{C}$

Explanation of Solution

Three charges are acting on the same line along the $x$-axis.

The figure for the position of point charges is shown below,

Figure (2)

According to Pythagoras theorem, the distance between the charge $q_1$ and position $Q\left(0,2.00\text{ m}\right)$

    $\begin{array}{c}{d}_1=\sqrt{{\left(0.50\text{ m}\right)}^2+{\left(2.00\text{ m}\right)}^2}\\ =2.06155\text{ m}\end{array}$

Here, $d_1$ is the distance between the charge $q_1$ and point $Q$

According to Pythagoras theorem, the distance between the charge $q_3$ and position $Q\left(0,2.00\text{ m}\right)$

    $\begin{array}{c}{d}_3=\sqrt{{\left(0.80\text{ m}\right)}^2+{\left(2.00\text{ m}\right)}^2}\\ =2.154\text{ m}\end{array}$

Here, $d_3$ is the distance between the charge $q_3$ and point $Q$.

Write the expression for the electric field due to charge $q_1$ at $Q$

    $\overrightarrow{E_1}=k_e\frac{q_1}{d_1{}^2}{\widehat{d}}_1$

Here, $E_1$ is the electric field due to charge $q_1$.

Substitute $9\times {10}^9{\text{Nm}}^2/{\text{C}}^2$ for $k_e$, $-4.00\text{ nC}$ for $q_1$ and $2.06155\text{ m}$ for $d_1$,

    $\begin{array}{c}\overrightarrow{E_1}=\frac{\left(9\times {10}^9{\text{Nm}}^2/{\text{C}}^2\right)\left(-4.00\text{ nC}\right)\left(\frac{1\text{C}}{{10}^9\text{ nC}}\right)}{{\left(2.06155\text{ m}\right)}^2}\widehat{d}\\ =-8.47\widehat{d}\text{N}/\text{C}\end{array}$

In component form the electric field can be written as,

    $\overrightarrow{E_1}=\left(-8.47\cos \theta \widehat{i}-8.47\sin \theta \widehat{\text{j}}\right)\text{N}/\text{C}$

According to right angle triangle property,

    $\begin{array}{c}\cos \theta =\frac{0.50}{2.06}\\ =0.2427\end{array}$

And,

    $\begin{array}{c}\sin \theta =\frac{2.00}{2.06}\\ =0.9708\end{array}$

Substitute $0.2427$ for $\cos \theta$ and $0.9708$ for $\sin \theta$ according to right angle triangle property to get $\overrightarrow{E_1}$,

    $\begin{array}{c}\overrightarrow{E_1}=\left(-8.47\times 0.2427\right)\widehat{\text{i}}-\left(8.47\times 0.9708\right)\widehat{\text{j}}\text{N}/\text{C}\\ =-2.05\widehat{\text{i}}-8.22\widehat{\text{j}}\text{N}/\text{C}\end{array}$

Write the expression for the electric field due to charge $q_2$ at $Q$

    $\overrightarrow{E_2}=k_e\frac{q_2}{d_2{}^2}{\widehat{d}}_2$

Here, $E_2$ is the electric field due to charge $q_2$ and $d_2$ is the distance between the charge $q_2$ and point $Q$

Substitute $9\times {10}^9{\text{Nm}}^2/{\text{C}}^2$ for $k_e$, $5.00\text{ nC}$ for $q_2$ and $2.00\text{ m}$ for $d_2$,

  $\begin{array}{c}\overrightarrow{E_2}=\frac{\left(9\times {10}^9{\text{Nm}}^2/{\text{C}}^2\right)\left(5.00\text{ nC}\right)\left(\frac{1\text{C}}{{10}^9\text{ nC}}\right)}{{\left(2.00\text{ m}\right)}^2}\widehat{d}\\ =11.25\widehat{\text{j}}\text{N}/\text{C}\end{array}$

Write the expression for the electric field due to charge $q_3$ at $Q$

    $\overrightarrow{E_3}=k_e\frac{q_3}{d_3{}^2}{\widehat{d}}_3$

Here, $E_3$ is the electric field due to charge $q_3$.

Substitute $9\times {10}^9{\text{Nm}}^2/{\text{C}}^2$ for $k_e$, $3.00\text{ nC}$ for $q_3$ and $2.154\text{ m}$ for $d_3$,

    $\begin{array}{c}\overrightarrow{E_3}=\frac{\left(9\times {10}^9{\text{Nm}}^2/{\text{C}}^2\right)\left(3.00\text{ nC}\right)\left(\frac{1\text{C}}{{10}^9\text{ nC}}\right)}{{\left(2.154\text{ m}\right)}^2}\widehat{d}\\ =5.819\widehat{d}\text{N}/\text{C}\end{array}$

In component form the electric field can be written as,

    $\overrightarrow{E_3}=\left(-5.819\cos \varphi \right)\widehat{\text{i}}+\left(5.819\sin \varphi \right)\widehat{\text{j}}\text{N}/\text{C}$

According to right angle triangle property,

    $\begin{array}{c}\cos \varphi =\frac{0.80}{2.154}\\ =0.371\end{array}$

And,

    $\begin{array}{c}\sin \varphi =\frac{2.00}{2.154}\\ =0.928\end{array}$

Substitute $0.371$ for $\cos \varphi$ and $0.928$ for $\sin \varphi$ according to right angle triangle property to get $\overrightarrow{E_3}$,

    $\begin{array}{c}\overrightarrow{E_3}=\left(-5.819\times 0.371\right)\widehat{\text{i}}+\left(5.819\times 0.928\right)\widehat{\text{j}}\text{N}/\text{C}\\ =-2.16\widehat{\text{i}}+5.40\widehat{\text{j}}\text{N}/\text{C}\end{array}$

Write the expression for the resultant electric field

    $\overrightarrow{E}=\overrightarrow{E_1}+\overrightarrow{E_2}+\overrightarrow{E_3}$

Here, $E$ is the resultant electric field

Conclusion:

Substitute $-2.05\widehat{\text{i}}-8.22\widehat{\text{j}}\text{N}/\text{C}$ for $\overrightarrow{E_1}$, $11.25\widehat{\text{j}}\text{N}/\text{C}$ for $\overrightarrow{E_2}$ and $-2.16\widehat{\text{i}}+5.40\widehat{\text{j}}\text{N}/\text{C}$ for $\overrightarrow{E_3}$,

    $\begin{array}{c}\overrightarrow{E}=\left(-2.05\widehat{\text{i}}-8.22\widehat{\text{j}}+11.25\widehat{\text{j}}-2.16\widehat{\text{i}}+5.40\widehat{\text{j}}\right)\text{ N}/\text{C}\\ =\left(-4.21\right)\widehat{\text{i}}+\left(8.43\right)\widehat{\text{j}}\text{ N}/\text{C}\end{array}$

Therefore, the electric field at position $\left(0,\text{ 2}\text{.00 m}\right)$ is $\left(-4.21\right)\widehat{\text{i}}+\left(8.43\right)\widehat{\text{j}}\text{ N}/\text{C}$