Question 16.1: Charge q1 is fixed at the origin, and q2 is free to move. (a...
Charge q_{1} is fixed at the origin, and q_{2} is free to move. (a) Suppose both charges are positive. If charge q_{2} is released from rest on the +x -axis, describe its subsequent motion. Tell what happens to the kinetic energy K, potential energy U, and total energy E. (b) Repeat part (a) assuming the charges q_{1} and q_{2} have opposite signs.
Coulomb’s law gives the force, and Equation 16.1 the potential energy. Coulomb’s law says that like charges repel, and unlike charges attract. Equation 16.1, U=k q_{1} q_{2} / r, holds whether the charges are like or unlike. In a conservative system, the total energy E = K + U is constant.
U=\frac{k q_{1} q_{2}}{r} (Potential energy of two point charges; SI unit: J) (16.1).
(a) The charges repel, so q_{1} feels a force to the right, and it accelerates in that direction (Figure 16.2a). Initially, the kinetic energy K is zero. The potential energy U is initially positive, because the product q_{1} q_{2} is positive. So the total energy E is positive, and it remains constant even after q_{1} starts moving. But as q_{1} accelerates, its kinetic energy becomes increasingly positive, and its potential energy drops so that E remains constant. This makes sense, because U=k q_{1} q_{2} / r, \text { and } r \text { is increasing as } q_{2} \text { moves away from } q_{1} .
(b) The opposite charges attract. With q_{1} \text { at the origin, } q_{2}, feels a force to the left, so it accelerates leftward (Figure 16.2b). Again, q_{2} \text { 's } kinetic energy K is initially zero, but now its potential energy U is negative, because the product q_{1} q_{2} is negative. So the total energy E is negative, and it remains constant even after q_{2} \text { starts moving. But again } q_{2}{ }‘s K increases as it accelerates, so U must become even more negative to keep E constant. Is this consistent with the separation r decreasing? Yes: As r decreases, U=k q_{1} q_{2} / rbecomes more negative. So U decreases as K increases, and the total energy E remains constant.
REFLECT Conservation of energy holds for the electric force -a conservative force. Potential energy can be positive or negative, depending on whether the charges are like or unlike.
