# Lecture 29. PHYC 161 Fall 2016

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1 Lecture 29 PHYC 161 Fall 2016

2 Magnetic Force and Torque on a Current Loop Let s look at the Net force and net torque on a current loop: df Idl B F IaB top and bottom F IbB sides But, the forces on opposite sides are opposing, so: F Net 0 Take the axis of rotation to be the y-axis, then the torque is: b b IaB sin IaB sin 2 2 IabB sin IABsin

3 Magnetic Torque on a Current Loop We can rewrite this as: IABsin Bsin or B Where the direction of the magnetic moment, µ, is given by the right-hand rule.

4 Force and torque on a current loop The net force on a current loop in a uniform magnetic field is zero. We can define a magnetic moment µ with magnitude IA, and direction as shown. The net torque on the loop is given by the vector product:

5 Magnetic dipole in a nonuniform magnetic field A current loop with magnetic moment pointing to the left is in a magnetic field that decreases in magnitude to the right. When these forces are summed to find the net force on the loop, the radial components cancel so that the net force is to the right, away from the magnet.

6 CPS 28-2 A circular loop of wire carries a constant current. If the loop is placed in a region of uniform magnetic field, the net magnetic force on the loop is A. perpendicular to the plane of the loop, in a direction given by a right-hand rule. B. perpendicular to the plane of the loop, in a direction given by a left-hand rule. C. in the same plane as the loop. D. zero. E. The answer depends on the magnitude and direction of the current and on the magnitude and direction of the magnetic field.

7 CPS 28-3 A circular loop of wire carries a constant current. If the loop is placed in a region of uniform magnetic field, the net magnetic torque on the loop A. tends to orient the loop so that its plane is perpendicular to the direction of the magnetic field. B. tends to orient the loop so that its plane is edge-on to the direction of the magnetic field. C. tends to make the loop rotate around its axis. D. is zero. E. The answer depends on the magnitude and direction of the current and on the magnitude and direction of the magnetic field.

8 The magnetic field of a moving charge A moving charge generates a magnetic field that depends on the velocity of the charge, and the distance from the charge.

9 Magnetic field of a straight current-carrying wire

10 Magnetic field of a current element The total magnetic field of several moving charges is the vector sum of each field. The magnetic field caused by a short segment of a currentcarrying conductor is found using the law of Biot and Savart:

11 Magnetic field of a straight current-carrying conductor Let s use the law of Biot and Savart to find the magnetic field produced by a straight currentcarrying conductor. The figure shows such a conductor with length 2a carrying a current I. We will find at a point a distance x from the conductor on its perpendicular bisector.

12 Magnetic field of a straight current-carrying conductor Since the direction of the magnetic field from all parts of the wire is the same, we can integrate the magnitude of the magnetic field and obtain: As the length of the wire approaches infinity, x >> a, and the distance x may be replaced with r to indicate this is a radius of a circle centered on the conductor:

13 Magnetic field of a straight current-carrying conductor The field lines around a long, straight, currentcarrying conductor are circles, with directions determined by the righthand rule.

14 Another HW problem

15

16 HW32

17 Magnetic field of a circular current loop Shown is a circular conductor with radius a carrying a counterclockwise current I. We wish to calculate the magnetic field on the axis of the loop.

18 Magnetic field of a circular current loop The magnetic field along the axis of a loop of radius a carrying a current I is given by the equation below. The direction is given by the right-hand rule shown.

19 Magnetic field lines of a circular current loop The figure shows some of the magnetic field lines surrounding a circular current loop (magnetic dipole) in planes through the axis. The field lines for the circular current loop are closed curves that encircle the conductor; they are not circles, however.

20 HW

21 Currents and planetary magnetism The earth s magnetic field is caused by currents circulating within its molten, conducting interior. These currents are stirred by our planet s relatively rapid spin (one rotation per 24 hours). The moon s internal currents are much weaker; it is much smaller than the earth, has a predominantly solid interior, and spins slowly (one rotation per 27.3 days). Hence the moon s magnetic field is only about 10 4 as strong as that of the earth.

22 Magnetic fields of current-carrying wires Computer cables, or cables for audio-video equipment, create little or no magnetic field. This is because within each cable, closely spaced wires carry current in both directions along the length of the cable. The magnetic fields from these opposing currents cancel each other.

23 Force between parallel conductors The magnetic field of the lower wire exerts an attractive force on the upper wire. If the wires had currents in opposite directions, they would repel each other.

24 Force between parallel conductors The figure shows segments of two long, straight, parallel conductors separated by a distance r and carrying currents I and I' in the same direction. Each conductor lies in the magnetic field set up by the other, so each experiences a force.

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