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Chapter 27: Problem 48

The magnetic field inside a solenoid of circular cross section is given by \(\vec{B}=b t \hat{k},\) where \(b=2.1\) T/ms. At time \(t=0.40 \mu \mathrm{s},\) a proton is inside the solenoid at \(x=5.0 \mathrm{cm}, y=z=0,\) and is moving with velocity \(\vec{v}=4.87 \mathrm{Mm} / \mathrm{s}\). Find the electromagnetic force on the proton.

Short Answer

Expert verified
The electromagnetic force on the proton can be calculated by performing a cross product between the velocity vector of the proton and the magnetic field vector and then multiplying the result by the charge of the proton.

Step by step solution

01

Identify given values

Identify and note the given values from the exercise. The exercise provides the following values: \(b = 2.1\) T/ms, \(t = 0.40 \mu s\), the proton's initial position \((x, y, z) = (5.0 cm, 0, 0)\), and the proton's velocity \(\vec{v} = 4.87 Mm/s\).
02

Calculate the Magnetic Field

Calculate the magnetic field at a given time using the provided expression for the magnetic field, which is \(\vec{B} = b t \hat{k}\). Substituting the given values, we have \(\vec{B} = (2.1 T/ms * 0.40 \mu s) \hat{k}\). This calculation gives the magnetic field vector at time \(t = 0.40 \mu s\).
03

Calculate the Lorentz force

Use the equation for the Lorentz force to calculate the force on a moving charged particle in a magnetic field. The Lorentz force \(\vec{F}\) is given by \(\vec{F} = q (\vec{v} \times \vec{B})\), where \(q\) is the charge of the proton with a value of \(1.6 * 10^{-19} C\), \(\vec{v}\) is the velocity of the proton, and \(\vec{B}\) is the magnetic field calculated in Step 2.
04

Calculate the Force

Insert the values into the Lorentz force equation to find the force. The velocity of proton \(\vec{v}\) has been given only in \(x\)-direction, which means its vector form would be \(\vec{v} = 4.87 Mm/s \hat{i}\), and \(\vec{B}\) has been calculated in previous step. Note as well that the given velocity in Mm/s should be converted to m/s before performing the calculation. As the force is a result of vector cross product, the result would be a force vector \(\vec{F} = q(\vec{v} \times \vec{B})\) acting on the proton.

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Most popular questions from this chapter

The current in a series \(R L\) circuit increases to \(20 \%\) of its final value in \(3.1 \mu \mathrm{s}\). If \(L=1.8 \mathrm{mH},\) what's the resistance?

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