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Chapter 27: Problem 8

An electron moves in a circular trajectory with radius \(r_{\mathrm{i}}\) in a constant magnetic field. What is the final radius of the trajectory when the magnetic field is doubled? a) \(\frac{r_{i}}{4}\) b) \(\frac{r_{i}}{2}\) c) \(r_{i}\) d) \(2 r_{i}\) e) \(4 r_{\mathrm{i}}\)

Short Answer

Expert verified
Question: When the magnetic field in which an electron is moving in a circular trajectory is doubled, the final radius of the trajectory is: a) Twice the initial radius b) Half the initial radius c) The same as the initial radius d) Four times the initial radius Answer: b) Half the initial radius

Step by step solution

01

Setting the magnetic force and centripetal force equal to each other

We have two equations: \(F = qvB\) (magnetic force) and \(F = \frac{mv^2}{r}\) (centripetal force). Since the magnetic force is providing the centripetal force for the electron, we can set these two equations equal to each other: $$ \Rightarrow qvB = \frac{mv^2}{r} $$
02

Solving for the radius in terms of the magnetic field

Now, we can solve for the radius \(r\) in terms of the magnetic field \(B\). Rearrange the equation as follows: $$ \Rightarrow r = \frac{mv}{qB} $$
03

Doubling the magnetic field

We are given that the magnetic field is doubled. Let \(r_i\) be the initial radius and \(r_f\) be the final radius. We can write two equations: $$ \begin{cases}r_i = \frac{mv}{qB_i}\\ r_f = \frac{mv}{q(2B_i)} \end{cases} $$
04

Finding the relationship between initial and final radii

Now, to find the relationship between \(r_i\) and \(r_f\), we can divide the second equation by the first equation: $$ \frac{r_f}{r_i} = \frac{\frac{mv}{q(2B_i)}}{\frac{mv}{qB_i}} = \frac{B_i}{2B_i} = \frac{1}{2} $$
05

Solving for the final radius

We have found the relationship between the initial and final radii, so we can now solve for the final radius \(r_f\) in terms of the initial radius \(r_i\): $$ \Rightarrow r_f = \frac{1}{2} r_i $$ The final radius of the trajectory when the magnetic field is doubled is \(\frac{r_i}{2}\). Therefore, the correct answer is option (b).

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Most popular questions from this chapter

In your laboratory, you set up an experiment with an electron gun that emits electrons with energy of \(7.50 \mathrm{keV}\) toward an atomic target. What deflection (magnitude and direction) would Earth's magnetic field \((0.300 \mathrm{G})\) produce in the beam of electrons if the beam is initially directed due east and covers a distance of \(1.00 \mathrm{~m}\) from the gun to the target? (Hint: First calculate the radius of curvature, and then determine how far away from a straight line the electron beam has deviated after \(1.00 \mathrm{~m}\).)

A particle with a charge of \(+10.0 \mu \mathrm{C}\) is moving at \(300 \cdot \mathrm{m} / \mathrm{s}\) in the positive \(z\) -direction. a) Find the minimum magnetic field required to keep it moving in a straight line at constant speed if there is a uniform electric field of magnitude \(100 . \mathrm{V} / \mathrm{m}\) pointing in the positive \(y\) -direction. b) Find the minimum magnetic field required to keep the particle moving in a straight line at constant speed if there is a uniform electric field of magnitude \(100 . \mathrm{V} / \mathrm{m}\) pointing in the positive \(z\) -direction.

An electron is moving at \(v=6.00 \cdot 10^{7} \mathrm{~m} / \mathrm{s}\) perpendicular to the Earth's magnetic field. If the field strength is \(0.500 \cdot 10^{-4} \mathrm{~T}\), what is the radius of the electron's circular path?

A square loop of wire of side length \(\ell\) lies in the \(x y\) -plane, with its center at the origin and its sides parallel to the \(x\) - and \(y\) -axes. It carries a current, \(i\), in the counterclockwise direction, as viewed looking down the \(z\) -axis from the positive direction. The loop is in a magnetic field given by \(\vec{B}=\left(B_{0} / a\right)(z \hat{x}+x \hat{z}),\) where \(B_{0}\) is a constant field strength, \(a\) is a constant with the dimension of length, and \(\hat{x}\) and \(\hat{z}\) are unit vectors in the positive \(x\) -direction and positive \(z\) -direction. Calculate the net force on the loop.

A proton moving at speed \(v=1.00 \cdot 10^{6} \mathrm{~m} / \mathrm{s}\) enters a region in space where a magnetic field given by \(\vec{B}=\) \((-0.500 \mathrm{~T}) \hat{z}\) exists. The velocity vector of the proton is at an angle \(\theta=60.0^{\circ}\) with respect to the positive \(z\) -axis. a) Analyze the motion of the proton and describe its trajectory (in qualitative terms only). b) Calculate the radius, \(r\), of the trajectory projected onto a plane perpendicular to the magnetic field (in the \(x y\) -plane). c) Calculate the period, \(T,\) and frequency, \(f\), of the motion in that plane. d) Calculate the pitch of the motion (the distance traveled by the proton in the direction of the magnetic field in 1 period).
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