Chapter 20, Problem 11STP

To determine

To Calculate: The net force exerted by charges A and B on charge C.

Answer to Problem 11STP

  $0.346\text{N}$ along the x-axis

Explanation of Solution

Given:

Charge on A, $q_A=-2.0\text{μC}$

Charge on B, $q_B=-2.0\text{μC}$

Charge on C, $q_C=-4.0\text{μC}$

Formula Used:

Electric force exerted on charge can be obtained by:

  $F_{xy}=k\frac{q_x{q}_y}{r_{xy}{}^2}$

Here, k is the Coulomb’s constant, $q_x$ and $q_y$ are the two charges and $r_{xy}$ is the distance between them.

Calculations:

Distance between charge A and C is:

  $\begin{array}{c}{r}_{AC}=\sqrt{{0.30}^2+{0.40}^2}\\ =\sqrt{0.25}\\ =0.5\text{m}\end{array}$

Force on charge C due to charge A can be calculated as follows:

  $\begin{array}{c}{F}_{AC}=\frac{k{q}_A{q}_C}{r_{AC}{}^2}\\ =\frac{9\times {10}^9\times \left(-2.0\times {10}^{-6}\right)\left(-4.0\times {10}^{-6}\right)}{{\left(0.50\right)}^2}\\ =0.288\text{N}\end{array}$

Angle with the horizontal at which this force acts:

  $\begin{array}{c}\tan {\theta }_{AC}=\frac{0.30}{0.40}\\ {\theta }_{AC}=36.9°\end{array}$

Distance between charge B and C is:

  $\begin{array}{c}{r}_{AC}=\sqrt{{0.30}^2+{0.40}^2}\\ =\sqrt{0.25}\\ =0.5\text{m}\end{array}$

Force on charge C due to charge B can be calculated as follows:

  $\begin{array}{c}{F}_{BC}=\frac{k{q}_B{q}_C}{r_{BC}{}^2}\\ =\frac{9\times {10}^9\times \left(-2.0\times {10}^{-6}\right)\left(-4.0\times {10}^{-6}\right)}{{\left(0.50\right)}^2}\\ =0.288\text{N}\end{array}$

Angle with the horizontal at which this force acts:

  $\begin{array}{c}\tan {\theta }_{BC}=\frac{0.30}{0.40}\\ {\theta }_{BC}=36.9°\end{array}$

  

The magnitude of the force due to charges A and B are equal but forces are in different direction. The sine components cancel leaving cosine sines in the same direction which add up giving the resultant force on charge C as:

  $\begin{array}{c}{F}_{net}=F_{AC}\sin {\theta }_{AC}+F_{BC}\sin {\theta }_{BC}\\ =0.288\sin 36.9°+0.288\sin 36.9°\\ =0.346\text{N}\end{array}$

Conclusion:

Thus, the net force exerted by charges A and B on charge C is $0.346\text{N}$ along the x-axis.