First slide

Potential energy

Question

Two charges q₁ and q₂ are placed 30 cm apart, as shown in the figure. A third charge q₃ is moved along the arc of a circle of radius 40 cm from C to D. The change in the potential energy of the system is $\frac{q_{3}}{4 {πε}_{0}} k$ , where k is

Moderate

A

8q₂
B

8q₁
C

6q₂
D

6q₁

Solution

$\begin{array}{rcl} u_{1} & = & \frac{1}{4 {πε}_{0}} [\frac{q_{1} q_{2}}{30 x 10^{- 2}} + \frac{q_{1} q_{3}}{40 x 10^{- 2}} + \frac{q_{2} q_{3}}{50 x 10^{- 2}}] \\ u_{2} & = & \frac{1}{4 {πε}_{0}} [\frac{q_{1} q_{2}}{30 x 10^{- 2}} + \frac{q_{1} q_{3}}{40 x 10^{- 2}} + \frac{q_{2} q_{3}}{10 x 10^{- 2}}] \\ ∆ u & = & u_{2} - u_{1} = \frac{q_{2} q_{3}}{4 {πε}_{0}} x 10^{2} [\frac{1}{10} - \frac{1}{50}] \\ = & \frac{q_{2} q_{3}}{4 {πε}_{0}} x \frac{4}{50} x 100 = \frac{q_{3}}{4 {πε}_{0}} (8 q_{2}) ∴ k = 8 q_{2} \end{array}$

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