Beam of particles in a cylindrical pipe

[CAF123]

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.

 

[tiny-tim]

Hi CAF123!

…with n particles per unit volume.
…
I then said that the volume charge density is $Q/\pi a^2 h$. So in the Gaussian cylinder, the charge enclosed is $(\pi r^2 h) \cdot Q/\pi a^2 h = \left(\frac{r}{a}\right)^2 nq$…

I don't understand …

q is charge, n is 1/volume

 

[CAF123]

Hi CAF123! I don't understand …

q is charge, n is 1/volume

Yes, there are n particles per unit volume so charge of nq per unit volume. So (volume) charge density is $nq/(\pi a^2 h)$. Then I multipled this by the volume of the Gaussian cylinder to get the charge within the Gaussian cylinder.

 

[tiny-tim]

Yes, there are n particles per unit volume so charge of nq per unit volume. So (volume) charge density is $nq/(\pi a^2 h)$.

No, there's a charge of nq per m³.

Volume of cylinder = πa²h m³, so total charge in cylinder = πa²hnq,
and charge inside radius r = πr²hnq