1. A wire carrying a current I is shaped as shown below. Find the magnitude of the magnetic field at point P using the Law of Biot-Savart. The identity òdx[x² + b²]^-3/2 = x/b²[x²+b²]^1/2 + C may be of assistance.



2. A wire carrying a current I is shaped as shown below. The arcs are circular of radii a and b where a > b. The straight pieces are radial from the centre of the shape. Each large arc subtends an angle of 2π /3. Find the magnitude of the magnetic field at the centre of the shape using the Law of Biot-Savart. The relationship S = rθ may be of use.



3. A loop of wire has the shape of two concentric semicircles connected by two radial segments. The loop carries current I as shown. Find the magnetic field at the point P using the Law of Biot-Savart.



4. An arc of thin wire of radius R lies in the positive quadrant of the xy plane. It carries a current I clockwise. Determine the integral expression for the magnetic field acting at x = a. Don't bother evaluating the integral.



5. A flat ribbon of length L and width W lies in the yz plane as shown below. It carries surface current K = kCz. Determine the magnetic field P = (b, 0, 0).



6. A semi–circular ribbon has inner radius R and outer radius R + a. The ribbon carries a surface current K = Csφ where s is radial distance and φ is the angular unit vector in polar coordinates. Determine the magnetic field acting at the centre of the curvature.



7. A flat disk of radius R lies in the xy plane as shown below. It carries surface charge density Σ . The disk is given angular velocity Ω φ where φ is the cylindrical rotational unit vector. Since the charge is moving, we now have a surface current density K = Σ sΩ φ where s is the distance from the centre of the disk. Determine the magnetic field acting at P = (0, 0, b).



8. A sphere of radius R is shown below. It carries volume charge density ρ . The sphere is given angular velocity Ω φ about the z axis where φ is the spherical rotational unit vector. Since the charge is moving, we now have a volume current density J = ρ rsin(θ )Ω φ where r is the radial distance from the centre of the sphere and angle θ is measured from the z-axis. Determine the magnetic field acting at P = (0, 0 , b).



9. A cylinder has its central axis oriented in the z direction. Its base is in the xy plane. The base has radius R and infinite height. It carries a current density J = Csz, where s is the cylindrical radial distance. Determine the magnetic field acting at a distance b from the centre of the base.
10. Which of the following are possible magnetic fields?
1. B_x = 2xy, B_y = 2z+3, and B_z = 5–2yz.
2. B_x = x²+y², B_y = 3, and B_z = –xz.
3. B_x = y³–z³, B_y = z³–x³, and B_z = x³–y³ .
4. B_x = ln(x), B_y = y/x, and B_z = 2y/x.

11. The diagrams below show a current-carrying loop of radius a located in the xz plane and a point P located in the yz plane. As shown in Figure 1, point P is a distance R from the centre of the loop at an angle α to the y axis. As shown in figure 2, point A is a point on the loop at an angle β from the z axis. Vector r = P - A is the distance from points A to P.
1. Obtain expressions for P, A, r, and dl in Cartesian (ijk) coordinates.
2. Find dl ´ r in Cartesian coordinates.
3. Use the Law of Biot-Savart to obtain integral expressions for the Cartesian components of the magnetic field at point P.
4. Assume R >> a and use the identity (1 – x)^z = 1 + zx for small x to expand and simplify the integrands in part (c). Integrate to show



where m = Iπ a² is the dipole moment of the loop.

5. Figure 3 shows the relationship between the k and j axes and the r and θ axes. Use this relationship to show that

Questions?mike.coombes@kwantlen.ca