The force on a current-carrying wire

1. The force on a current-carrying wire A magnetic field exerts a force on a single moving charge, so it's not surprising that it exerts a force on a current-carrying wire, seeing as a current is a set of moving charges. Using q = It, this becomes: But a velocity multiplied by a time is a length L, so this can be written: The direction of the force is given by the right-hand rule, where your fingers point in the direction of the current. Current is defined to be the direction of flow of positive charges, so your right hand always gives the correct direction.

2. The right-hand rule • A wire carries current into the page in a magnetic field directed down the page. In which direction is the force? • 1. Left • 2. Right • 3. Up • 4. Down • 5. Into the page • 6. Out of the page • 7. The net force is zero

3. Three wires Consider three wires carrying identical currents between two points, a and b. The wires are exposed to a uniform magnetic field. Wire 1 goes directly from a to b. Wire 2 consists of two straight sections, one parallel to the magnetic field and one perpendicular to the field. Wire 3 takes a meandering path from a to b. Which wire experiences more force? 1. Wire 1 2. Wire 2 3. Wire 3 4. equal for all three

4. Three wires The force is equal for all three. What matters is the displacement perpendicular to the field, and that's equal for all wires carrying equal currents between the same two points in a uniform magnetic field.

5. The force on a current-carrying loop • A wire loop carries a clockwise current in a uniform magnetic field directed into the page. In what direction is the net force on the loop? • 1. Left • 2. Right • 3. Up • 4. Down • 5. Into the page • 6. Out of the page • 7. The net force is zero

6. The force on a current-carrying loop The net force is always zero on a current-carrying loop in a UNIFORM magnetic field.

7. Is there a net anything on the loop? Let’s change the direction of the uniform magnetic field. Is the net force on the loop still zero? Is there a net anything on the loop?

8. Is there a net anything on the loop? Let’s change the direction of the uniform magnetic field. Is the net force on the loop still zero? Is there a net anything on the loop? The net force is still zero, but there is a net torque that tends to make the loop spin.

9. The torque on a current loop The magnetic field is in the plane of the loop and parallel to two sides. If the loop has a width a, a height b, and a current I, then the force on each of the left and right sides is F = IbB. The other sides experience no force because the field is parallel to the current in those sides. Simulation The torque ( ) about an axis running through the center of the loop is:

10. The torque on a current loop ab is the area of the loop, so the torque here is . This is the maximum possible torque, when the field is in the plane of the loop. When the field is perpendicular to the loop the torque is zero. In general, the torque is given by: where q is the angle between the area vector, A, (which is perpendicular to the plane of the loop) and the magnetic field, B.

11. A DC motor A direct current (DC) motor is one application of the torque exerted on a current loop by a magnetic field. The motor converts electrical energy into mechanical energy. If the current always went the same way around the loop, the torque would be clockwise for half a revolution and counter-clockwise during the other half. To keep the torque (and the rotation) going the same way, a DC motor usually has a "split-ring commutator" that reverses the current every half rotation. Simulation

12. Producing a magnetic field Electric fields are produced by charges. Magnetic fields are produced by moving charges. In practice, we generally produce magnetic fields from currents.

13. The magnetic field from a long straight wire The long straight current-carrying wire, for magnetism, is analogous to the point charge for electric fields. The magnetic field a distance r from a wire with current I is: , the permeability of free space, is:

14. The magnetic field from a long straight wire Magnetic field lines from a long straight current-carrying wire are circular loops centered on the wire. The direction is given by another right-hand rule. Point your right thumb in the direction of the current (out of the screen in the diagram, and the fingers on your right hand, when you curl them, show the field direction.

15. The force between two wires • A long-straight wire carries current out of the page. A second wire, to the right of the first, carries current into the page. In which direction is the force that the second wire feels because of the first wire? • 1. Left • 2. Right • 3. Up • 4. Down • 5. Into the page • 6. Out of the page • 7. The net force is zero

16. The force between two wires In this situation, opposites repel and likes attract! Parallel currents going the same direction attract. If they are in opposite directions they repel.

17. The net magnetic field • In which direction is the net magnetic field at the origin in the situation shown below? All the wires are the same distance from the origin. • 1. Left • 2. Right • 3. Up • 4. Down • 5. Into the page • 6. Out of the page • 7. The net field is zero

18. The net magnetic field We add the individual fields to find the net field, which is directed right.

19. A loop and a wire • A loop with a clockwise current is placed below a long straight wire carrying a current to the right. In which direction is the net force exerted by the wire on the loop? • 1. Left • 2. Right • 3. Up • 4. Down • 5. Into the page • 6. Out of the page • 7. The net force is zero

20. A loop and a wire The long straight wire creates a non-uniform magnetic field, pictured below.

21. A loop and a wire The forces on the left and right sides cancel, but the forces on the top and bottom only partly cancel – the net force is directed up, toward the long straight wire.

22. A loop and a wire I1 a The forces on the left and right sides cancel, but the forces on the top and bottom only partly cancel – the net force is directed up, toward the long straight wire. I2 b L

23. Five wires Four long parallel wires carrying equal currents perpendicular to your page pass through the corners of a square drawn on the page, with one wire passing through each corner. You get to decide whether the current in each wire is directed into the page or out of the page. First we’ll have a fifth parallel wire, carrying current into the page, that passes through the center of the square. Can you choose current directions for the other four wires so that the fifth wire experiences a net force directed toward the top right corner of the square?

24. How many ways? • You can choose the direction of the currents at each corner. How many configurations give a net force on the center wire that is directed toward the top-right corner? • 1. 1 • 2. 2 • 3. 3 • 4. 4 • 5. 0 or more than 4

25. How many ways? Note: if the length of each side is d, and the currents are all I, the net force per unit length here is: First, think about the four forces we need to add to get a net force toward the top right. How many ways can we create this set of four forces?

26. How many ways? How many ways can we create this set of four forces? Two. Wires 1 and 3 have to have the currents shown. Wires 2 and 4 have to match, so they either both attract or both repel. Currents going the same way attract; opposite currents repel.

27. Four wires Now we’ll remove the fifth wire and focus on the net magnetic field at the center of the square because of the other four wires. Can you choose current directions for the four wires so that the net magnetic field at the center is directed toward the top right corner of the square?

28. How many ways? • You can choose the direction of the currents at each corner. How many configurations give a net magnetic field at the center that is directed toward the top-right corner? • 1. 1 • 2. 2 • 3. 3 • 4. 4 • 5. 0 or more than 4

29. How many ways? Note: if the length of each side is d, and the currents are all I, the net field is: First, think about the four fields we need to add to get a net field toward the top right. How many ways can we create this set of four fields?

30. How many ways? How many ways can we create this set of four fields? Two. Wires 2 and 4 have to have the currents shown. Wires 1 and 3 have to match, so their fields cancel. The right-hand rule: Point your thumb in the direction of the current, and your curled fingers show the direction of the field.

31. The field from a solenoid A solenoid is simply a coil of wire with a current going through it. It's basically a bunch of loops stacked up. Inside the coil, the field is very uniform (not to mention essentially identical to the field from a bar magnet). For a solenoid of length L, current I, and total number of turns N, the magnetic field inside the solenoid is given by:

32. The field from a solenoid We can make this simpler by using n = N/L as the number of turns per unit length, to get: . The magnetic field is almost uniform - the solenoid is the magnetic equivalent of the parallel-plate capacitor. If we put a piece of ferromagnetic material (like iron or steel) inside the solenoid, we can magnify the magnetic field by a large factor (like 1000 or so).

33. A bar magnet and a solenoid A bar magnet field looks like the field of a solenoid. Why?

34. A bar magnet and a solenoid A bar magnet field looks like the field of a solenoid. Why? The currents associated with the atoms mostly cancel inside the bar magnet, but they add together around the outside, giving something that looks remarkably like a solenoid.