The Timpson Lyceum · University Physics II (2026) – Electricity & Magnetism

Chapter 28 — Sources of Magnetic Field

Problem sheet · 13 points · 3 extra credit

Chapter 28 — Sources of Magnetic Field

Core: 13 points · Extra credit: 3 points. Show your reasoning; partial credit is given.

1. 1. (2 pts ••) A point charge $q=+2.0\,\text{nC}$ moves with speed $v=4.0\times10^{6}\,\text{m/s}$. Consider a field point a distance $r=3.0\,\text{mm}$ from the charge.

   (a) Find the magnitude of the magnetic field this charge produces at that point when the point lies in the direction perpendicular to the velocity.

   (b) Find the field magnitude at the same distance for a point whose position vector makes a $30^\circ$ angle with the velocity.

2. 2. (2 pts ••) A long, straight wire carries a steady current $I=8.0\,\text{A}$.

   (a) Find the magnetic field magnitude at a perpendicular distance $r=5.0\,\text{cm}$ from the wire.

   (b) At what perpendicular distance does the field drop to $20\,\mu\text{T}$?

3. 3. (2 pts ••) Two long, straight, parallel wires are separated by $d=10\,\text{cm}$. Each carries a current of $6.0\,\text{A}$, but in opposite directions. Find the magnitude of the net magnetic field at the point midway between the wires, and state the reasoning that lets you add the two contributions.

4. 4. (2 pts ••) A flat circular coil has $N=50$ turns, radius $R=4.0\,\text{cm}$, and carries a current $I=2.5\,\text{A}$.

   (a) Find the magnetic field magnitude at the center of the coil.

   (b) Find the field magnitude on the coil's axis at a distance $x=3.0\,\text{cm}$ from the center.

5. 5. (2 pts ••) Two long, straight, parallel wires are $2.0\,\text{cm}$ apart. One carries $I_1=10\,\text{A}$ and the other $I_2=15\,\text{A}$ in the same direction.

   (a) Find the force per unit length each wire exerts on the other.

   (b) State whether the wires attract or repel, and why.

6. 6. (1 pt •) A solenoid is $40\,\text{cm}$ long and is wound with $1200$ turns of wire carrying a current of $3.0\,\text{A}$. Treating it as an ideal (long) solenoid, find the magnetic field magnitude inside it.

7. 7. (2 pts ••) A toroid has $N=500$ turns and carries a current $I=4.0\,\text{A}$. A point inside the toroid's core lies a distance $r=10\,\text{cm}$ from the central axis.

   (a) Use Ampère's law to find the magnetic field magnitude at that point.

   (b) Explain, using the same Ampèrian-loop argument, why the field is zero in the hole at the very center of the toroid (on the central axis).

8. Extra Credit (3 pts •••)

   A single circular loop of radius $R$ carries a steady current $I$. Using the Biot–Savart law, derive the magnetic field on the axis of the loop at a distance $x$ from the center.

   (a) Set up the Biot–Savart integral for a current element $I\,d\vec{\ell}$, and explain carefully why only the axial component of $d\vec B$ survives when you integrate around the full loop.

   (b) Carry out the integration to obtain

   $$B_x=\frac{\mu_0 I R^2}{2\,(R^2+x^2)^{3/2}} .$$

   (c) Show that your result reduces to the correct field at the center of the loop, and find its numerical value for $R=4.0\,\text{cm}$, $I=2.5\,\text{A}$, at the point $x=R$.