Sources of Magnetic Field: Summary

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1 Sources of Magnetic Field: Summary Single Moving Charge (Biot-Savart for a charge): Steady Current in a Wire (Biot-Savart for current): Infinite Straight Wire: Direction is from the Right Hand Rule

2 The Magnetic Field of a Current Loop Your author uses the Biot-Savart Law to find the magnetic field on the axis of a current loop: On the Axis: Direction is by the RHR: fingers wrap around in direction of current, thumb points in direction of the field.

3 Whiteboard Problem: 32-8 At what distance on the axis of a current loop is the magnetic field half the strength of the field at the center of the loop? Give your answer as a multiple of R, i.e. some number times R. (LC)

4 The Dipole Field As you showed in the lab activity, both a bar magnet and a current loop have a dipole field: (PhET Faraday Lab)

5 Dipole Moment of a Current Loop Using the equation for the field on the axis of a loop for z >> R: Where the magnetic dipole moment is defined by: (watch the different mu s!) Important Point: A magnetic dipole moment behaves in an external magnetic field just like an electric dipole moment in an external electric field.

6 Whiteboard Problem: 32-9 The Earth s magnetic dipole moment is 8.0 X A m 2. a) What is the magnetic field strength on the surface of the Earth at the Earth s north magnetic pole? You can assume that the current loop is deep inside the Earth. (LC) Look up a value for the magnetic field strength at the surface of the Earth; is your result close? b) Astronauts discover an Earth sized planet without a magnetic field. To create one with the same strength as the Earth s, they propose running a current through a wire around the equator. What size current would be needed? (LC)

7 Introduction: Ampere s Law We re at the same place in our study of the magnetic field as we were with the electric field at the end of Chapter 26: Recall what Gauss said for the Electric Field: Of course, as we saw in Chapter 27, Gauss Law is a concise mathematical statement of this. For any closed 2D surface S, the electric field will be such that:

8 Ampere s Law Ampere noticed something similar for the magnetic field and currents: Ampere s Law: General Statement For any currents and any closed 1D path in space: Note that we have to choose a positive direction for the current: a current one way will produce a field in one direction on the path, and an opposite current will produce a field the other way.

9 Whiteboard Problem Solution: Use Ampere s Law with current coming out as positive: (LC) (Is this a mean problem or what?)

10 How Do We Use Ampere s Law To Find the Magnetic Field? Recall that Gauss Law contains the same physics as Coulomb s Law, but it uses a different language the language of fields. Ampere s Law is similar; it contains the same physics as the Biot-Savart Law, but it too is in the language of fields. How can we use Ampere s Law to find the magnetic field? Ampere s Law is always true for any distribution of currents and any path; however, it is only useful in finding the field if there is enough symmetry to easily determine the line integral i.e. do it in your head! Here s an example of something we already know: an infinite straight wire: Amperian Path (integration path) Side view By symmetry: Same thing we got using the Biot-Savart Law.

11 Whiteboard Problem 32-11: An Infinitely Long Wire of Radius R An infinitely long wire of radius R carries current I assumed to be uniform over the cross section of the wire. Part 1: Find the magnitude of the magnetic field outside the wire for r > R: (LC) Cross Section Diagram: Amperian path Since all of the current goes through the path, the steps are the same as for a thin wire. Answer: as if all of the current is concentrated on the axis

12 Whiteboard Problem : An Infinitely Long Wire of Radius R Part 2: Find the magnitude of the magnetic field inside the wire for r < R: (LC) Cross Section Diagram: Amperian path Answer:

13 Whiteboard Problem : An Infinitely Long Wire of Radius R Part 3: Plot the magnitude of the magnetic field as a function of r: (LC)

14 A Short Pause: Maxwell s Equations As I m sure you remember me saying, one of our goals in our study of electricity and magnetism is to arrive at Maxwell s Equations, a complete description of all classical Electricity and Magnetism. So far, we ve talked about one, but we have some more: Maxwell Equation #1: Maxwell Equation #2: If charges create the electric field, and we know there are no isolated magnetic charges (poles): (PhET Faraday Lab) Maxwell Equation #3: There s only one more Maxwell Equation to go!

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