a) What is the electric potential energy at a point midway between two charges, -2 `mu`C and -4 `mu`C, seperated by 20 cm?
b) What is the electric potential energy stored in the combination described in part a?
a) Electric potential energy is not defined for a point in space; it is defined for a charged object or a combination of charges. However, we can calculate electric potential of the point midway between the two given charges. According to the superposition principle, the electric potential at this point is the sum of the electric potentials due to each charge:
`U = k q_1/r_1 + k q_2/r_2` , where `r_1 ` and `r_2` are the distances between the point and each charge. In this case since the point is midway between the two charges separated by 20 cm, `r_1=r_2=r = 10 cm = 0.1m` .
In this formula, k is the constant `k=9*10^9 (N*m^2)/C^2` and the charges `q_1 =-2 muC=-2*10^-6 C` and `q_2 = -4 muC = -4*10^-6 C` .
Plugging these values in the formula for U, we get
`U=` `k q_1/r + k q_2/r = k/r (q_1 + q_2) = (9*10^9)/0.1 * (-2 - 4)*10^-6 V = 540*10^3 V= 5.4*10^5V`
The electric potential is `5.4*10^5 ` Volts.
b) The electrical potential energy stored in this combination is the potential energy of interaction between the charges:
`E_(el) = k(q_1 *q_2)/r_12` , where `r_12` is the distance between the charges, 20 cm = 0.2 m
`E_(el) = 9*10^9 * ((-2*10^-6)(-4*10^-6))/0.2 J = 0.36 J`
The electric potential energy stored in the combination of charges is 0.36 J.