- #1

dsilvas

- 7

- 4

- Homework Statement
- What is the length L of the longest drift tube in a linac, which, operating at a frequency f = 20 MHz, is capable of accelerating Carbon-12 ions to a maximum energy of E = 100 MeV?

- Relevant Equations
- (1): E = ½mv^2

(2): L = v/2f

**Clarification:**

The statement in the title is actually from the solution to the homework question, as given by the textbook (you can see the whole thing below under "Textbook solution"). The solution doesn't explain everything, which is where my confusion comes from. Usually in my classes we ignore relativistic effects when we know v << c, but in this case it says to ignore relativistic effects because E << m where m is the rest mass of a carbon 12 ion. I don't understand why you can do that.

**Numbers/Unit conversion for rest mass and energy:**

The max energy in the problem is given to be 100 MeV. I looked up the mass of the Carbon 12 ion and it was m = 1.9927e-26 kg. The energy can be converted from 100 MeV to be

E = 1.6021773e-11 Joules = 1.6021773e-11 kg*m

^{2}/s

^{2}

**What I'm confused about:**

One of the problems I'm having is the energy in this case is clearly much GREATER than the rest mass, which is the exact opposite of the condition stated in the solution. The other problem I'm having is understanding where this condition came from in the first place.

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**How to get the solution:**

It looks like we are supposed to end up using equation (1) (which is justified by saying E << m somehow), which ends up getting the desired result of v = 4.01e7, which we can then plug into equation (2) to get the length L of the drift tube. My final answer for L is 1.00251143 meters, and the textbook solution just rounds it to 1 meter which is fair. The final answer itself isn't the trouble I'm having.

**Textbook solution:**

"For constant acceleration, the ions must travel the length of the drift tube in half a cycle of the r.f. field. Thus, L = υ/2 f , where υ is the velocity of the ion. Since the energy is far less than the rest mass of the ion, we can use nonrelativistic kinematics to find υ, i.e. υ = 4.01 × 10

^{7}m s−1 and finally L = 1 m. "

**Final remark:**

Sorry if my formatting isn't that great, this is my first time on this website and I'm sure using code makes things easier to read, but I don't really know how to use that yet. Hope this works!