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Question: Three point charges Q1, Q2 and Q3 are arranged as shown in the figure. The work that needs to be don...

Three point charges Q1, Q2 and Q3 are arranged as shown in the figure. The work that needs to be done by an external agent to separate the charges to a large distance from each other very slowly will be

Answer

4.65 x 10^{-3} J

Explanation

Solution

To find the work done by an external agent to separate the charges to a large distance, we need to calculate the negative of the total electrostatic potential energy of the initial configuration of the charges. The final potential energy, when charges are at infinite separation, is zero.

The electrostatic potential energy of a system of point charges is the sum of the potential energies of all unique pairs of charges. The potential energy between two point charges qiq_i and qjq_j separated by a distance rijr_{ij} is given by: Uij=kqiqjrijU_{ij} = k \frac{q_i q_j}{r_{ij}} where k=9×109 N m2/C2k = 9 \times 10^9 \text{ N m}^2/\text{C}^2.

From the figure, we have the following charges and distances: Q1=+1.0μC=+1.0×10−6 CQ_1 = +1.0 \mu\text{C} = +1.0 \times 10^{-6} \text{ C} Q2=−1.0μC=−1.0×10−6 CQ_2 = -1.0 \mu\text{C} = -1.0 \times 10^{-6} \text{ C} Q3=−2.0μC=−2.0×10−6 CQ_3 = -2.0 \mu\text{C} = -2.0 \times 10^{-6} \text{ C}

The distances between the charges are: r12=4 mr_{12} = 4 \text{ m} (between Q1Q_1 and Q2Q_2) r13=3 mr_{13} = 3 \text{ m} (between Q1Q_1 and Q3Q_3) r23=5 mr_{23} = 5 \text{ m} (between Q2Q_2 and Q3Q_3)

Now, we calculate the potential energy for each pair:

  1. Potential energy between Q1Q_1 and Q2Q_2 (U12U_{12}): U12=kQ1Q2r12U_{12} = k \frac{Q_1 Q_2}{r_{12}} U12=(9×109 N m2/C2)(+1.0×10−6 C)(−1.0×10−6 C)4 mU_{12} = (9 \times 10^9 \text{ N m}^2/\text{C}^2) \frac{(+1.0 \times 10^{-6} \text{ C})(-1.0 \times 10^{-6} \text{ C})}{4 \text{ m}} U12=(9×109)−1.0×10−124 JU_{12} = (9 \times 10^9) \frac{-1.0 \times 10^{-12}}{4} \text{ J} U12=−94×10−3 JU_{12} = - \frac{9}{4} \times 10^{-3} \text{ J} U12=−2.25×10−3 JU_{12} = -2.25 \times 10^{-3} \text{ J}

  2. Potential energy between Q1Q_1 and Q3Q_3 (U13U_{13}): U13=kQ1Q3r13U_{13} = k \frac{Q_1 Q_3}{r_{13}} U13=(9×109 N m2/C2)(+1.0×10−6 C)(−2.0×10−6 C)3 mU_{13} = (9 \times 10^9 \text{ N m}^2/\text{C}^2) \frac{(+1.0 \times 10^{-6} \text{ C})(-2.0 \times 10^{-6} \text{ C})}{3 \text{ m}} U13=(9×109)−2.0×10−123 JU_{13} = (9 \times 10^9) \frac{-2.0 \times 10^{-12}}{3} \text{ J} U13=−183×10−3 JU_{13} = - \frac{18}{3} \times 10^{-3} \text{ J} U13=−6.0×10−3 JU_{13} = -6.0 \times 10^{-3} \text{ J}

  3. Potential energy between Q2Q_2 and Q3Q_3 (U23U_{23}): U23=kQ2Q3r23U_{23} = k \frac{Q_2 Q_3}{r_{23}} U23=(9×109 N m2/C2)(−1.0×10−6 C)(−2.0×10−6 C)5 mU_{23} = (9 \times 10^9 \text{ N m}^2/\text{C}^2) \frac{(-1.0 \times 10^{-6} \text{ C})(-2.0 \times 10^{-6} \text{ C})}{5 \text{ m}} U23=(9×109)+2.0×10−125 JU_{23} = (9 \times 10^9) \frac{+2.0 \times 10^{-12}}{5} \text{ J} U23=+185×10−3 JU_{23} = + \frac{18}{5} \times 10^{-3} \text{ J} U23=+3.6×10−3 JU_{23} = +3.6 \times 10^{-3} \text{ J}

The total electrostatic potential energy of the system (UtotalU_{\text{total}}) is the sum of these pairwise energies: Utotal=U12+U13+U23U_{\text{total}} = U_{12} + U_{13} + U_{23} Utotal=(−2.25×10−3 J)+(−6.0×10−3 J)+(+3.6×10−3 J)U_{\text{total}} = (-2.25 \times 10^{-3} \text{ J}) + (-6.0 \times 10^{-3} \text{ J}) + (+3.6 \times 10^{-3} \text{ J}) Utotal=(−2.25−6.0+3.6)×10−3 JU_{\text{total}} = (-2.25 - 6.0 + 3.6) \times 10^{-3} \text{ J} Utotal=(−8.25+3.6)×10−3 JU_{\text{total}} = (-8.25 + 3.6) \times 10^{-3} \text{ J} Utotal=−4.65×10−3 JU_{\text{total}} = -4.65 \times 10^{-3} \text{ J}

The work done by an external agent (WextW_{\text{ext}}) to separate the charges to a large distance (where the final potential energy is zero) very slowly is equal to the negative of the initial potential energy of the system: Wext=−UtotalW_{\text{ext}} = -U_{\text{total}} Wext=−(−4.65×10−3 J)W_{\text{ext}} = -(-4.65 \times 10^{-3} \text{ J}) Wext=+4.65×10−3 JW_{\text{ext}} = +4.65 \times 10^{-3} \text{ J}

This can also be expressed as 4.65 mJ4.65 \text{ mJ}.