Work, Energy, and Emf — Question 6

Question 6

Emf is the line integral of the non-electrostatic force per charge

Along a source channel $0<x<L$, the non-electrostatic force on positive charge q is $\vec F_{\mathrm{ne}}=qf_0(1+x/L)\hat x$. Traversal is quasistatic and internal losses are negligible.

1. Find the emf for travel from x=0 to x=L.

2. Find the average opposing electrostatic field component and the net electric work over an external return path.

Question 6 - Solution

Emf is the line integral of the non-electrostatic force per charge

(a) Method and calculation

$$\mathcal E=\int_0^L(\vec F_{\mathrm{ne}}/q)\cdot d\vec\ell=f_0\int_0^L(1+x/L)dx.$$$$\mathcal E=\tfrac32f_0L,\quad V(L)-V(0)=\mathcal E.$$

(b) Interpretation and further calculation

$$\langle E_x\rangle=-\frac{V(L)-V(0)}L=-\tfrac32f_0.$$$$W_{E,\mathrm{return}}=q\mathcal E=\tfrac32qf_0L.$$

Physical check

The source force depends on its energy-conversion mechanism; its work is not part of the conservative electrostatic loop integral.