Electric Charge — Question 7

Question 7

A global reversal of charge signs leaves pair forces unchanged

Four fixed point charges have arbitrary nonzero signed values $q_i$. At fixed geometry, assume the force from pair $i,j$ is a geometric vector factor times $q_iq_j$. Compare two operations: reverse all four charges, or reverse only $q_1$.

1. Derive the effect of global reversal on every pair force and on the net force of each particle.

2. If the original force on particle 1 is $(2\hat{\mathbf i}-5\hat{\mathbf j})$ mN, find it after reversal of $q_1$ alone. Determine whether net forces on particles 2 through 4 must also reverse.

Question 7 - Solution

A global reversal of charge signs leaves pair forces unchanged

(a) Method and calculation

$$\mathbf F_{ij}=\mathbf G_{ij}q_iq_j,\quad(-q_i)(-q_j)=q_iq_j.$$$$\mathbf F_{ij}\prime=\mathbf F_{ij},\quad\mathbf F_i\prime=\sum_{j\ne i}\mathbf F_{ij}=\mathbf F_i.$$

(b) Interpretation and further calculation

$$\mathbf F_1\prime=-\mathbf F_1=(-2\hat{\mathbf i}+5\hat{\mathbf j})\,\mathrm{mN}.$$$$i\ne1:\quad\mathbf F_i\prime=\mathbf F_i-2\mathbf F_{i1}.$$For each other particle only the contribution from particle 1 reverses. Its whole resultant generally does not.

Physical check

The symmetry statement applies to electrostatic forces; an external force not reversed with the charges would not necessarily share it.