Chapter 27 — Magnetic Field & Magnetic Forces

14 points · 3 extra credit

Chapter 27 — Magnetic Field & Magnetic Forces

Core: 14 points · Extra credit: 3 points. Show your reasoning; partial credit is given.

1. 1. (2 pts ••) A proton moves with speed $v=2.5\times10^{5}\,\text{m/s}$ through a uniform magnetic field of magnitude $B=0.85\,\text{T}$. Its velocity makes an angle of $30^\circ$ with the field.

   (a) Find the magnitude of the magnetic force on the proton.

   (b) The velocity points in the $+x$ direction and the field lies in the $xy$-plane, tilted $30^\circ$ above the $x$-axis toward $+y$. In which direction does the force on the proton point? Justify using the right-hand rule.

   (c) If the particle were an electron with the same velocity, how would the force direction change?

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2. 2. (3 pts •••) A charged particle enters a region of uniform magnetic field with its velocity perpendicular to the field and travels in a circle.

   (a) Starting from $\mathbf{F}=q\mathbf{v}\times\mathbf{B}$ and Newton's second law, derive the expression for the radius $r=mv/qB$ of the circular path.

   (b) An electron moves at $v=3.0\times10^{6}\,\text{m/s}$ perpendicular to a field $B=0.45\,\text{T}$. Find the radius of its orbit.

   (c) Find the period of the motion and the cyclotron frequency $f=qB/2\pi m$. Confirm numerically that these do not depend on the speed.

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3. 3. (2 pts ••) A velocity selector uses crossed electric and magnetic fields so that only particles with one speed pass straight through undeflected. The magnetic field is $B=0.12\,\text{T}$, and singly charged ions of speed $v=4.0\times10^{5}\,\text{m/s}$ are to pass through.

   (a) What electric field magnitude is required, and how must $\mathbf{E}$ and $\mathbf{B}$ be oriented relative to $\mathbf{v}$?

   (b) The field is produced by two parallel plates separated by $1.5\,\text{cm}$. What voltage must be applied across them?

   (c) Explain what happens to an ion moving faster than the selected speed.

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4. 4. (2 pts ••) A straight wire of length $L=0.35\,\text{m}$ carries a current $I=8.5\,\text{A}$ in a uniform magnetic field $B=0.60\,\text{T}$.

   (a) Find the force on the wire when the current is perpendicular to $\mathbf{B}$.

   (b) Find the force when the wire makes an angle of $55^\circ$ with $\mathbf{B}$.

   (c) At what angle is the force zero?

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5. 5. (2 pts ••) A flat rectangular coil of $N=50$ turns, dimensions $4.0\,\text{cm}\times6.0\,\text{cm}$, carries a current $I=1.2\,\text{A}$ in a uniform field $B=0.30\,\text{T}$.

   (a) Find the magnetic dipole moment $\mu=NIA$ of the coil.

   (b) Find the maximum torque the field can exert on the coil.

   (c) Find the work required to rotate the coil from the orientation of minimum potential energy (moment aligned with $\mathbf{B}$) to the orientation where the moment is antiparallel to $\mathbf{B}$.

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6. 6. (2 pts ••) A copper ribbon $0.50\,\text{mm}$ thick and $2.0\,\text{cm}$ wide carries a current $I=25\,\text{A}$ perpendicular to a magnetic field $B=0.80\,\text{T}$ directed through the flat face of the ribbon. The carrier density of copper is $n=8.5\times10^{28}\,\text{m}^{-3}$.

   (a) Find the Hall voltage measured across the width of the ribbon.

   (b) Explain how the sign of the Hall voltage reveals whether the charge carriers are positive or negative.

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7. 7. (1 pt •) An electron enters a uniform field $B=0.30\,\text{T}$ with speed $v=5.0\times10^{5}\,\text{m/s}$, its velocity making an angle of $25^\circ$ with the field, so it follows a helix.

   (a) Find the radius of the helix.

   (b) Find the pitch (the distance advanced along the field per revolution).

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8. Extra Credit (3 pts •••)

   A particle of charge $q$ and mass $m$ starts from rest at the origin in a region of crossed static fields: a uniform electric field $\mathbf{E}=E\,\hat{\mathbf{y}}$ and a uniform magnetic field $\mathbf{B}=B\,\hat{\mathbf{z}}$.

   (a) Write the equations of motion for $v_x(t)$ and $v_y(t)$ from $\mathbf{F}=q(\mathbf{E}+\mathbf{v}\times\mathbf{B})$, and show they are coupled first-order equations.

   (b) Solve them with the initial condition $\mathbf{v}(0)=0$. (Hint: define the cyclotron frequency $\omega=qB/m$ and look for the steady drift plus an oscillation.)

   (c) Integrate to find $x(t)$ and $y(t)$, and show the trajectory is a cycloid with a constant drift velocity $v_d=E/B$ in the $+x$ direction. Sketch the path and give the maximum displacement of the particle in the $y$-direction.

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