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A charge q in a uniform electric field Eo experiences a
Chapter 4, Problem 4.17(choose chapter or problem)
A charge q in a uniform electric field Eo experiences a constant force F = q Eo. (a) Show that this force is conservative and verify that the potential energy of the charge at position r is U (r) = qE0 r. (b) By doing the necessary derivatives, check that F = V U .
Questions & Answers
QUESTION:
A charge q in a uniform electric field Eo experiences a constant force F = q Eo. (a) Show that this force is conservative and verify that the potential energy of the charge at position r is U (r) = qE0 r. (b) By doing the necessary derivatives, check that F = V U .
ANSWER:Step 1 of 3
a) The force \(\mathbf{F} = q\mathbf{E}_o\) certainly doesn't depend on anything but \(\mathbf{r}\) (and doesn't even depend on \(\mathbf{r}\) since \(\mathbf{E}_o\) is constant).
The work integral is \(W(1\rightarrow 2)=\int_{1}^{2}\mathbf{F}.d\mathbf{r}=q\mathbf{E}_o \cdot \int_{1}^{2}d\mathbf{r}=q\mathbf{E}_o\cdot (\mathbf{r}_2-\mathbf{r}_1)\), which is independent of path.