Potential Energy of a System of Multiple Point Charges

Q. Charges –q, Q and –q are placed at an equal distance on a straight line. If the total potential energy of the system of three charges is zero, then find the ratio $\frac{Q}{q}$.

Q. Three point charges q, -2q and -2q are placed at the vertices of an equilateral triangle of side a. The work done by some external force to increase their separation to 2a will be

1. $\frac{1}{4\pi {ϵ}_0}.\frac{2q^2}{a}$
2. $Negative$
3. $Zero$
4. $\frac{1}{4\pi {ϵ}_0}.\frac{3q^2}{a}$

Q. Three charges each of charge ‘q’ are placed at the corners of an equilateral triangle. The fourth charge to be placed at the centre of the triangle so that electrostatic potential energy of the system is zero is

1. $\frac{q}{\sqrt{3}}$
2. $\frac{-q}{\sqrt{3}}$
3. $q$
4. $\frac{-q}{\sqrt{2}}$

Q. An infinite number of charges, each numerically equal to $q$ and of the same sign, are placed along the x-axis at $x=1\text{m},x=2\text{m},x=4\text{m},x=8\text{m}$ and so on. Then the electrical potential at $x=0$ due to this set of charges is ( Assume that $\frac{1}{4\pi {ϵ}_0}=k$)

1. $2kq$
2. $\frac{2kq}{5}$
3. $\frac{4kq}{3}$
4. $\frac{4kq}{5}$

Q.

Six point charges, each of +1 C, are brought together to the vertices of regular hexagon from infinity one by one. Assuming infinity as the reference point, what is the change in potential energy of the system?
Take the length of each side of the hexagon to be 1m.

1. Positive
2. Negative
3. Zero
4. None of these

Q. Three point charges q, -2q and -2q are placed at the vertices of an equilateral triangle of side a. The work done by some external force to increase their separation to 2a will be

1. $\frac{1}{4\pi {ϵ}_0}.\frac{2q^2}{a}$
2. $Negative$
3. $Zero$
4. $\frac{1}{4\pi {ϵ}_0}.\frac{3q^2}{a}$

Q. Four charges q, -q, q and -q are placed at the vertices A, B, C and D of a square of side a. Find the work done in interchanging charges at B and C.

1. zero
2. $\frac{k{q}^2}{a}(4-\sqrt{2})$
3. $\frac{k{q}^2}{a}(4-2\sqrt{2})$
4. $\frac{k{q}^2}{2a}(4-\sqrt{2})$

Q. An infinite number of charges, each numerically equal to $q$ and of the same sign, are placed along the x-axis at $x=1\text{m},x=2\text{m},x=4\text{m},x=8\text{m}$ and so on. Then the electrical potential at $x=0$ due to this set of charges is ( Assume that $\frac{1}{4\pi {ϵ}_0}=k$)

1. $2kq$
2. $\frac{2kq}{5}$
3. $\frac{4kq}{3}$
4. $\frac{4kq}{5}$

Q.

There are 3 charges of +1C, +2C and +3C at the vertices of an equilateral triangle of length 2m. What is the Potential energy of the system assuming potential energy at infinite separation to be zero?

1. $4.95\times {10}^{10}J$

2. $5.95\times {10}^{10}J$

3. $2.95\times {10}^{10}J$

4. $3.95\times {10}^{10}J$

Q. Two Charges $Q_1$ and $Q_2$ are as shown in the figure. A third charge $Q_3$ is moved from points A to B along a circular path. Change in potential energy of the system is

1. $k{Q}_1{Q}_2{Q}_3$
2. $4k{Q}_1{Q}_2$
3. $4k{Q}_2{Q}_3$
4. $\frac{2}{3}k{Q}_2{Q}_3$

Q. Two charges $q_1$ and $q_2$ are placed $30\text{cm}$ apart, as shown in the figure. A third charge $q_3$ is moved along the arc of a circle of radius $40\text{cm}$ from $C$ to $D$. The change in the potential energy of the system is $\frac{q_3}{4\pi {ϵ}_0}k$, where $k$ is

1. $8q_2$
2. $6q_2$
3. $8q_1$
4. $6q_1$

Q. Two charges each $+5\mu C$ are at $(0,\pm 3m)$. A third charge of $1\mu C$ and of mass 2g is at (4m, 0). What is the minimum velocity to be given to the $1\mu C$ charge such that it just reaches the origin (in $m{s}^{-1}$).

1. 3
2. 12
3. $\sqrt{12}$
4. 4

Q. Three particles, each of charge $10\mu C$ are placed at the corners of an equilateral triangle of side 10 cm. The electrostatic potential energy of the system is
Given$(\frac{1}{4\pi ϵ}=9\times {10}^{9}N-m^2/C^2)$

1. Zero
2. Infinite
3. 27 J
4. 100 J

Q. Two charges each $+5\mu C$ are at $(0,\pm 3m)$. A third charge of $1\mu C$ and of mass 2g is at (4m, 0). What is the minimum velocity to be given to the $1\mu C$ charge such that it just reaches the origin (in $m{s}^{-1}$).

1. 3
2. 12
3. $\sqrt{12}$
4. 4

Q. Three charges Q, +q and +q are placed at the vertices of a right angled isosceles triangle as shown. The net electrostatic energy of the configuration is zero if Q is equal to (IIT-JEE 2000)

1. $\frac{-q}{1+\sqrt{2}}$
2. $\frac{-2q}{2+\sqrt{2}}$
3. -2q
4. +q

Q. Four charges q, -q, q and -q are placed at the vertices A, B, C and D of a square of side a. Find the work done in interchanging charges at B and C.

1. zero
2. $\frac{k{q}^2}{a}(4-\sqrt{2})$
3. $\frac{k{q}^2}{a}(4-2\sqrt{2})$
4. $\frac{k{q}^2}{2a}(4-\sqrt{2})$

Q. Two charges $q_1$ and $q_2$ are located $30\text{cm}$ apart, as shown in the figure. A third charge $q_3$ is moved along tthe arc of a circle of radius $40\text{cm}$ from $C$ to $D$. The change in the potential energy of the system is $\frac{n{q}_2{q}_3}{4\pi {ϵ}_0}$. Find $n$.

Q. Three charges A, B and C, which were initially at infinite separation from each other, are brought together slowly by an external agent to form an equilateral triangle ABC. The experiment is repeated twice such that in the first case B is brought first under the influence of A followed by C. In the second case C is brought firstunder the influence of A followed by B. Work done by external agent is greater in the first case.

1. True
2. False

Q. Two Charges $Q_1$ and $Q_2$ are as shown in the figure. A third charge $Q_3$ is moved from points A to B along a circular path. Change in potential energy of the system is

1. $k{Q}_1{Q}_2{Q}_3$
2. $4k{Q}_1{Q}_2$
3. $4k{Q}_2{Q}_3$
4. $\frac{2}{3}k{Q}_2{Q}_3$