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Chapter 39: Problem 50

(a) By what factor must the magnetic field in a proton synchrotron be increased as the proton energy increases by a factor of \(10 ?\) Assume the protons are highly relativistic, so \(\gamma \gg 1\). (b) By what factor must the diameter of the accelerator be increased to raise the energy by a factor of 10 without changing the magnetic field?

Short Answer

Expert verified
The magnetic field and the diameter of the accelerator must both be increased by a factor of 10 to raise the energy by a factor of 10.

Step by step solution

01

Identify and Write Down the Relevant Formula

The relevant formula for this problem is the equation for the magnetic rigidity, \( Bρ = γmv \), where \( B \) is the magnetic field, \( ρ \) is the radius of curvature of the particle's path (which is half the diameter of the accelerator), \( γ \) is the Lorentz factor, \( m \) is the mass of the particle and \( v \) is its velocity. Since the protons are highly relativistic, we know that \( γ \gg 1 \) and therefore, \( v \approx c \), the speed of light. Thus, the equation simplifies to \( Bρ = γmc \).
02

Find the Factor by Which the Magnetic Field Must Be Increased

We want to find out by what factor the magnetic field must be increased as the proton energy increases by a factor of 10. The equation \( E = γmc^2 \) relates energy and the Lorentz factor. Thus, when the energy increases by a factor of 10, γ also increases by a factor of 10. So in \( Bρ = γmc \), when \( γ \) increases by a factor of 10, \( B \) must also increase by a factor of 10, assuming \( ρ \) remains constant.
03

Find the Factor by Which the Diameter of the Accelerator Must Be Increased

We now need to find out by what factor the diameter of the accelerator (\( 2ρ \)) must be increased to raise the energy by a factor of 10 without changing the magnetic field. We will use the same equation \( Bρ = γmc \) and the fact that \( E = γmc^2 \), so when energy (\( E \)) increases by a factor of 10, \( γ \) also increases by a factor of 10. Because \( B \) remains constant in this case, \( ρ (and therefore diameter) \) must also increase by a factor of 10.

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