CAF123
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Homework Statement
Charged particles, each holding charge q are moving in a cylinderical beam centred on the x-axis with n particles per unit volume. All the particles have the same horizontal velocity v.
A) By considering a suitable Gaussian surface, calculate the E-field as a function of r, the radial distance from the x-axis, and hence the force on the charges particle due to the electric field.
2. Homework Equations
Gauss Law,
The Attempt at a Solution
Let a be the radius of the pipe. Choose a Gaussian cylinder to be of radius r < a. Then the E field (from the enclosed charge) and the dA elements are parallel, so by Gauss,## E∫dA = Q_{enc}/ε = E(2 \pi r h),## h the height of the pipe and Gaussian cylinder.
I then said that the volume charge density is ##Q/\pi a^2 h##. So in the Gaussian cylinder, the charge enclosed is ##(\pi r^2 h) \cdot Q/\pi a^2 h = \left(\frac{r}{a}\right)^2 nq## which then gives me the E field and hence the force. My problem is, when I checked the solutions, they say the charge enclosed is ##Q = nq \pi r^2 h## and then they get an E field of ##nrq/2\epsilon##. To be honest, I think this is wrong. This expression for Q yields incorrect dimensions and then when they calculate the E field, they have ##Nm^3/C## which again is wrong. Both my expressions give the correct dimensions. Am I correct?
Many thanks.
