Energy problem taken from classical mechanics by john r taylor

Question

hi for the following problem i know the method to do it, im just struggling when it comes to evaluating the required methods as i havent seen one like this before,heres the problem

" a Charge $q$ in a uniform electric Field $E_0$ experiences a constant force $F=qE_0$" part a is to show that this force is conservative, to do this i need to show that the curl of $F$ is equal to 0 as my understanding is telling me The problem is actually calculating the Curl as i have no idea what to do with the $ E_0$ term, do i need to write it in a different form before i evaluate it directly?

The second part is verify that the potential energy of the charge at position $r$ is $U(r)=-qE_0.r$ again i'm having issues when evaluating the integral. if someone could show me how to evaluate these i would be extremly greatful.

this problem is 4.17 in the text book however the solutions in the back of the book is just for odd numbered questions

Answer 1

So $$ E_0 = (E_{01}, E_{02}, E_{03})^T = \text{const} $$ a vector of three constant real numbers and $$ F = q E_0 $$ thus that constant vector multiplied by a constant scalar $q$, giving a constant force vector $F$ and $$ \DeclareMathOperator{grad}{grad} \DeclareMathOperator{curl}{curl} (\curl F)_i = \epsilon_{ijk} \partial_j F_k = \epsilon_{ijk} \partial_j (q E_{0k}) = 0 $$ because a partial derivative of a constant is zero.

If $U(r) = -q E_0 \cdot r$ then \begin{align} F &= -\grad U \\ &= -\grad (-q E_0 \cdot r) \\ &= q \grad(E_0 \cdot r) \\ &= q \grad(E_{01}x_1 + E_{02}x_2 + E_{03} x_3) \\ &= q (E_{01} \grad x_1 + E_{02} \grad x_2 + E_{03} \grad x_3) \\ &= q (E_{01}(1,0,0)^T + E_{02}(0,1,0)^T + E_{03}(0,0,1)^T) \\ &= qE_0 \end{align} so we can recover the force, the potential fits up to some constant, meaning $U(r) + C$ would give that force as well. To single out that $U(r)$ the requirement $U(0) = 0$ must exist.

From the other direction: $$ dU = -dW = -F \cdot dr $$ gives $$ U(r) - U(0) = \int\limits_0^r dU \\ = - \int\limits_0^r F \cdot dr' \\ = - q \int\limits_0^r E_0 \cdot dr' \\ = - q E_0 \cdot \int\limits_0^r dr' \\ = -q E_0 \cdot (r - 0) \\ = -q E_0 \cdot r $$