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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 certain fixed length \(L\) of wire carries a current \(I\) (a) Show that if the wire is formed into a square coil, then the maximum torque in a given magnetic field \(B\) is developed when the coil has just one turn. (b) Show that the magnitude of this torque is \(\tau=\frac{1}{16} L^{2} I B\)
The conversion between atomic mass units and kilograms is $$1 \mathrm{u}=1.66 \times 10^{-27} \mathrm{kg}$$ A sample containing sulfur (atomic mass 32 u), manganese \((55 \mathrm{u}),\) and an unknown element is placed in a mass spectrometer. The ions have the same charge and are accelerated through the same potential difference before entering the magnetic field. The sulfur and manganese lines are separated by \(3.20 \mathrm{cm},\) and the unknown element makes a line between them that is \(1.07 \mathrm{cm}\) from the sulfur line. (a) What is the mass of the unknown clement? (b) Identify the element.
The strength of Earth's magnetic field, as measured on the surface, is approximately \(6.0 \times 10^{-5} \mathrm{T}\) at the poles and $3.0 \times 10^{-5} \mathrm{T}$ at the equator. Suppose an alien from outer space were at the North Pole with a single loop of wire of the same circumference as his space helmet. The diameter of his helmet is \(20.0 \mathrm{cm} .\) The space invader wishes to cancel Earth's magnetic field at his location. (a) What is the current required to produce a magnetic ficld (due to the current alone) at the center of his loop of the same size as that of Earth's field at the North Pole? (b) In what direction does the current circulate in the loop, CW or CCW, as viewed from above, if it is to cancel Earth's field?
A straight wire segment of length \(25 \mathrm{cm}\) carries a current of $33.0 \mathrm{A}$ and is immersed in a uniform external magnetic field. The magnetic force on the wire segment has magnitude \(4.12 \mathrm{N} .\) (a) What is the minimum possible magnitude of the magnetic field? (b) Explain why the given information enables you to calculate only the minimum possible field strength.
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