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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 * v²) / 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 * v²) / 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 * v²) / 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 × 10^(-27) kg, the charge (q) is 1.6 × 10^(-19) C, the maximum proton speed (v) is 1.0 × 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 × 10^(-27) kg * 1.0 × 10^(7) m/s) / (1.6 × 10^(-19) C * 0.50 T) R ≈ 0.104 m The minimum radius of the dees in the cyclotron must be approximately 0.104 m.

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Most popular questions from this chapter

A uniform magnetic field of \(0.50 \mathrm{T}\) is directed to the north. At some instant, a particle with charge \(+0.020 \mu \mathrm{C}\) is moving with velocity \(2.0 \mathrm{m} / \mathrm{s}\) in a direction \(30^{\circ}\) north of east. (a) What is the magnitude of the magnetic force on the charged particle? (b) What is the direction of the magnetic force?
A solenoid of length \(0.256 \mathrm{m}\) and radius \(2.0 \mathrm{cm}\) has 244 turns of wire. What is the magnitude of the magnetic field well inside the solenoid when there is a current of \(4.5 \mathrm{A}\) in the wire?
In an electric motor, a circular coil with 100 turns of radius $2.0 \mathrm{~cm}$ can rotate between the poles of a magnet. When the current through the coil is \(75 \mathrm{~mA},\) the maximum torque that the motor can deliver is \(0.0020 \mathrm{~N} \cdot \mathrm{m} .\) (a) What is the strength of the magnetic field? (b) Is the torque on the coil clockwise or counterclockwise as viewed from the front at the instant shown in the figure?
Two concentric circular wire loops in the same plane each carry a current. The larger loop has a current of 8.46 A circulating clockwise and has a radius of \(6.20 \mathrm{cm} .\) The smaller loop has a radius of \(4.42 \mathrm{cm} .\) What is the current in the smaller loop if the total magnetic field at the center of the system is zero? [See Eq. \((19-16) .]\)
An electromagnetic flowmeter is to be used to measure blood speed. A magnetic field of \(0.115 \mathrm{T}\) is applied across an artery of inner diameter \(3.80 \mathrm{mm}\) The Hall voltage is measured to be \(88.0 \mu \mathrm{V} .\) What is the average speed of the blood flowing in the artery?
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