Physics 1220 In-Class Problems: Law of Biot-Savart & Ampere's Law

1. Two wires are carrying conventional currents, I₁ = 30.0 A and I₂ = 22.0 A, in opposite directions. Determine the direction and magnitude of the net magnetic field at points A, B, and C where r₁ = 0.250 m, r₂ = 0.350 m, r₃ = 0.700 m, and r₄ = 0.500 m.

   

2. Determine the direction and magnitude of the net magnetic field at point A and B due to the two wires shown below. The wires carry conventional current I₁ = 650 mA and I₂ = 475 mA. Point A is a distance r = 1.20 m from wire 1 and point B is r₂ = 2.20 m away.

   

3. 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 magnetic field in component form at A? 
4. 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.
5. 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.
6. 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.
   

   

7. 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?
   

   

Law of Biot-Savart


8. 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 ∫dx[x²+b²]^-3/2 = x/(b²[x²+b²]^½ + C) may be of assistance.
   
9. 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.
   

10. 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.
    
11. A thin very long (treat as semi-infinite) wire carries a conventional current I as shown. Find the magnetic field at point P which is a distance a from the end of the wire.
    

12. The magnetic field at the centre of radius of curvature of a loop is a very common problem. In this problem, the arc has radius of curvature R that goes from zero to απ and the conventional current I is counter clockwise. Find and solve the integral expression for the magnetic field. The relationships S = Rθ may be of use and dℓ = dℓ [−i sinθ + j cosθ] may be helpful.
    

Ampere's Law

13. A long copper wire of cross-sectional radius R carries a uniform current I.  Use Ampere's Law to determine B as a function of the distance a from the centre of the wire. Sketch the result.
14. A coaxial cable consists of two concentric very thin, very long, cylindrical shells of radius R and 2R. The shells carry equal and opposite currents I.  Determine the magnetic field B as a function of the distance a from the centre of the wire.  Sketch the result.
15. 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. Use Ampere's Law to find the magnetic fields as a function of radial distance from the centre of the pipe. Sketch the result.
16. 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. Find formulas for the magnetic field in each of the regions 0 < a < r₁, r₁ < a < r₂, r₂ < a < r₃ , and a > r₃, where a is the radial distance from the centre of the cable. Sketch the result.
17. 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 ∫ e^-axxdx = -(x/a)e^-ax + e^-ax/a + C may be of use.
    

Questions? mike.coombes@kpu.ca