Homework for Spins

  1. (Euler)

    1. Use Euler's formula $e^{i\theta}=\cos\theta+i\sin\theta$ and its complex conjugate to find formulas for $\sin\theta$ and $\cos\theta$. In your physics career, you will often need to read these formula “backwards,” i.e. notice one of these combinations of exponentials in a sea of other symbols and say, “Ah ha! that is $\cos\theta$.” So, pay attention to the result of the homework problem!

    2. Show that $e^{2i\theta}+e^{-4i\theta}=2e^{-i\theta}\cos(3 \theta)$ by manipulating the left hand side until it looks like the right hand side of the equation. This calculation is similar to many calculations you will make in the course of your careers related to time or spatial dependence of quantum systems. So, pay attention to the methods of this homework problem!

  2. (Monopole)

    Using the formula $$\vec\mu=\frac{1}{2}\int\vec r\times \vec J(\vec r)\, d\tau$$ find the dipole moment of a wire carrying current $I$ and bent in the shape of a square with sides of length $L$.

  3. (MuDisk)

    Using the formula $$\vec\mu=\frac{1}{2}\int\vec r\times \vec J(\vec r)\, d\tau$$ find the dipole moment of a charged spinning disk with constant charge density, total charge $Q$, radius $R$, and period $T$.

  4. (ForceTorque)

    Consider a square wire loop with sides of length $L$ carrying current $I$. The normal to the plane of the wire loop is at an angle $\theta$ with respect to a uniform magnetic field $\vec B$. Take the direction of the magnetic field to be $\hat{z}$, the origin of coordinates to be at the center of the loop, the high side of the wire to be at constant positive values of $x$, and the current to be flowing counter-clockwise if looking down along the $z$-axis.

    1. Find the force on each side the wire loop due to the magnetic field.

      (Hint: For a current carrying wire, $d\vec{F=Id\vec{\ell} \times \vec{B}$})

    2. Find the net force on the loop. Consider the Physical Implication: What does this result mean for the motion of the loop? Compare \& Contrast Systems: How does this result compare/contrast with the example we did in class?

    3. Find the torque on each side of the wire loop due to the magnetic field.

    4. Find the net torque on the wire loop. Consider the Physical Implication: What does this result mean for the motion of the loop?

    5. Show that the (potential) energy $H$ of the wire loop in the external magnetic field is given by: $$H=-\vec{\mu}\cdot\vec{B}$$

      (Hint: To find the work done by a torque during a rotation, integrate the torque over the rotation angle.)

    6. Examine Special Cases: For what configuration of the loop and field would you expect the energy to be minimum? Maximum? Does the energy equation agree with your analysis?


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