Answer to Question #138239 in Electric Circuits for Kinder

Question #138239
Two particles, with charges of 20.0 nC and 40.0 nC are placed at the points with coordinates (0, 4.00 cm) and (3.00 cm, 0). A particle with charge 10.0 nC is located at the origin. (a) Find the electric potential energy of the configuration of the three fixed charges.
1
Expert's answer
2020-10-23T07:16:39-0400


Let 20.0 nC20.0\ nC charge be q1q_1, 40.0 nC40.0\ nC charge be q2q_2 and 10.0 nC10.0\ nC charge be q3q_3. Let, also, the distance between the charges q1q_1and q2q_2 be r12r_{12}, the distance between the charges q2q_2and q3q_3 be r23r_{23} and the distance between the charges q1q_1and q3q_3 be r13r_{13}.

Let's write the formula for the electric potential energy due to charges q1q_1and q2q_2:


U1=kq1q2r12=kq1q2(r132+r232),U_1=k\dfrac{q_1q_2}{r_{12}}=k\dfrac{q_1q_2}{\sqrt{(r_{13}^2+r_{23}^2)}},

here, k=8.99109 Nm2C2k=8.99\cdot 10^9 \ \dfrac{Nm^2}{C^2} is Coulomb constant, r13=4102 mr_{13}=4\cdot 10^{-2}\ m is the distance between the charges q1q_1and q3q_3, r23=3102 mr_{23}=3\cdot 10^{-2}\ m is the distance between the charges q2q_2and q3q_3.

Then, we can calculate U1U_1:

U1=8.99109 Nm2C220.0109 C40.0109 C((4102 m)2+(3102 m)2)=1.44104 J.U_1=8.99\cdot 10^9 \ \dfrac{Nm^2}{C^2}\cdot \dfrac{20.0\cdot 10^{-9}\ C \cdot 40.0\cdot 10^{-9}\ C}{\sqrt{((4\cdot 10^{-2}\ m)^2+(3\cdot 10^{-2}\ m)^2)}}=1.44\cdot 10^{-4}\ J.

Similarly, we can write the formula for the electric potential energy due to charges q2q_2and q3q_3:


U2=kq2q3r23.U_2=k\dfrac{q_2q_3}{r_{23}}.


Let's calculate U2U_2:

U2=8.99109 Nm2C240.0109 C10.0109 C3102 m=1.2104 J.U_2=8.99\cdot 10^9 \ \dfrac{Nm^2}{C^2}\cdot\dfrac{40.0\cdot 10^{-9}\ C \cdot 10.0\cdot 10^{-9}\ C}{3\cdot 10^{-2}\ m}=1.2\cdot 10^{-4}\ J.

Finally, we can write the formula for the electric potential energy due to charges q1q_1and q3q_3:


U3=kq1q3r13.U_3=k\dfrac{q_1q_3}{r_{13}}.


Let's calculate U3U_3:

U3=8.99109 Nm2C220.0109 C10.0109 C4102 m=0.45104 J.U_3=8.99\cdot 10^9 \ \dfrac{Nm^2}{C^2}\cdot\dfrac{20.0\cdot 10^{-9}\ C \cdot 10.0\cdot 10^{-9}\ C}{4\cdot 10^{-2}\ m}=0.45\cdot 10^{-4}\ J.

Then, the net electric potential energy of the configuration of the three fixed charges is the sum of U1U_1, U2U_2 and U3U_3:


Unet=U1+U2+U3,U_{net}=U_1+U_2+U_3,

Unet=1.44104 J+1.2104 J+0.45104 J=3.1104 J.U_{net}=1.44\cdot 10^{-4}\ J+1.2\cdot 10^{-4}\ J+0.45\cdot 10^{-4}\ J=3.1\cdot 10^{-4}\ J.

Answer:

Unet=3.1104 J.U_{net}=3.1\cdot 10^{-4}\ J.


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