/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 26 The magnetic field in a cyclotro... [FREE SOLUTION] | 91影视

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Chapter 19: Problem 26

The magnetic field in a cyclotron is \(0.50 \mathrm{T}\). What must be the minimum radius of the dees if the maximum proton speed desired is $1.0 \times 10^{7} \mathrm{m} / \mathrm{s} ?$

Short Answer

Expert verified
Answer: The minimum radius of the dees in the cyclotron must be approximately 0.104 m.

Step by step solution

01

Identify the relevant formula

The Lorentz force (F) acting on a charged particle of charge q moving with a velocity v in a magnetic field B is given by: F = q * (v x B) Where "x" denotes the cross product between velocity and magnetic field vectors. For a cyclotron, the charged particle moves in a circular path due to the magnetic field, so F also represents the centripetal force. Centripetal force is given by: F = (m * v虏) / R Where m is the mass of the particle, and R is the radius of the circular path.
02

Equate the two expressions for force

As the Lorentz force and the centripetal force both represent the force acting on the proton in a cyclotron, we can equate the two expressions: q * (v x B) = (m * v虏) / R
03

Simplify the equation

In a cyclotron, the magnetic field B is perpendicular to the velocity v. Therefore, the cross product simplifies to: v x B = v * B Now, we can substitute this back into the equation: q * (v * B) = (m * v虏) / R
04

Solve for the radius R

We can rearrange the equation to find the minimum radius of the dees (R): R = (m * v) / (q * B) For a proton, the mass (m) is 1.67 脳 10^(-27) kg, the charge (q) is 1.6 脳 10^(-19) C, the maximum proton speed (v) is 1.0 脳 10^(7) m/s, and the magnetic field (B) is 0.50 T. Using these values, we can now calculate R.
05

Calculate the minimum radius

Substitute the given values into the equation for R: R = (1.67 脳 10^(-27) kg * 1.0 脳 10^(7) m/s) / (1.6 脳 10^(-19) C * 0.50 T) R 鈮� 0.104 m The minimum radius of the dees in the cyclotron must be approximately 0.104 m.

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