Physics 1102 In-Class Problems: Law of Biot-Savart & Ampere's Law

1. A +1 μC charge is located at (0,0,0) and travelling with speed (1000,0,0). Another +2 μC charge is located at (3,4,0) and travelling with speed (5000,5000,0). Find the magnetic force that the electrons exert on one another.

2. 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]^½ + C) may be of assistance.
   

3. A wire carrying a current I is shaped as shown below. The arcs are circular of radii a and b. The straight pieces a radial to the centre of the shape. 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.
   
   

4. 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.
   
   

5. A +6.00 μC charge is moving with a speed of 7.50 × 10⁶ m/s parallel to a long, straight wire. The wire carries a current of 67.0 A in a direction opposite to that of the moving charge, and is 5.00 cm from the charge. Find the magnitude and direction of the force on the charge.
   

6. In the diagram below, the cross-section of two wires is shown. The wire on the left carries a current of I₁ = 12.5 A directed into the paper while the left has current I₂ = 8.5 A directed out of the paper. Both wires are R = 0.45 m from point A and the θ = 90°. What is the direction and magnitude of the magnetic field at A?
   

7. An electron is travelling 10.0 cm above and parallel to a long thin wire carrying 8.00 A of current. The conventional current is out of the paper as shown and the wire is parallel to the surface of the earth. What is the wire's magnetic field at the electron location? With what speed and direction must the electron be moving such that the magnetic force exactly balances the force due to gravity? The charge of an electron is -1.60 × 10^-19 C and its mass is 9.11 × 10^-31 kg.
   

8. Two rigid rods are oriented parallel to each other and to the ground. The rods carry the same current in the same direction. The length of each rod is 0.85 m, while the mass of each is 0.073 kg. One rod is held in place above the ground, and the other floats beneath it at a distance of 8.2 × 10^-3 m. Determine the current in the rods.
   

9. Two current carrying wires are parallel to each as shown in diagram (i) below. The side view is given in diagram (ii). The wires are 0.35 m apart. The current in the first wire is 25 A and 18 A in the second. The wires are 15.0 m long. What is the magnetic field (magnitude and direction) at the second wire due to the first wire? What is the force (magnitude and direction) on the second wire because of the magnetic field?
   

10. A long copper wire of cross-sectional radius R carries a uniform current I. What is the current density j(r)? Use Ampere's Law to determine B as a function of the distance a from the centre of the wire. Sketch the result.
    

11. A long copper pipe with thick walls has an inner radius R and an outer radius 2R. A current I flows along this wall, uniformly distributed over the cross-sectional area of the copper. What is the current density j(r) for all r? Use Ampere's Law to find the magnetic fields as a function of radial distance from the centre of the pipe. Sketch the result.
    

12. A coaxial cable consists of a long cylindrical copper wire of radius r₁ surrounded by a cylindrical insulating shell of outer radius r₂ . A final conducting cylindrical shell of outer radius r₃ surrounds the insulating shell. The wire and conducting shell carry equal but opposite currents I uniformly distributed over their volumes. What is the current density j(r) for all r? Find formulas for the magnetic field in each of the regions 0 < a < r₁, r₁ < a < r₂, r₂ < a < r₃ , and a > r₃. Sketch the result.

13. A long copper wire of cross-sectional radius R carries a current density j(r) = Ae^-Kr. Use Ampere's Law to determine B as a function of the distance a from the centre of the wire. Sketch the result. The integral identity òe^-axxdx = -(x/a)e^-ax + e^-ax/a + C may be of use.
    

Questions? mike.coombes@kpu.ca