Physics 1B Electricity & Magne4sm. Frank Wuerthwein (Prof) Edward Ronan (TA) UCSD

Transcription

1 Physics 1B Electricity & Magne4sm Frank Wuerthwein (Prof) Edward Ronan (TA) UCSD

2 Outline of today Con4nue Chapter 22 Magne4c field Solenoid Torque on a current loop

3 Magne4c Fields We have found the magne4c field from a long, straight current- carrying wire. But magne4c fields can be caused by any shape of current- carrying wire or any moving electrical charge. Take a curved section of currentcarrying wire: Notice how the center of the curved wire now has a straight magnetic field (like a bar magnet).

4 Magne4c Fields Let s examine a circle of wire that contains a current, I. Use RHR2 to find the magne4c field in the middle of the circular loop. " Take the magnetic field contribution from a small piece of wire on the right side of the wire. " In the middle of the loop, the magnetic field would point out of the page (board). " Now looking at a small piece of wire from the left, we again find that the magnetic field points out of the page (board).

5 Magne4c Fields All of the small sec4ons of wire will contribute to enhance the magne4c field at the center of the wire. " Forming the wire into a loop strengthens the magnetic field inside the loop compared to a straight wire. " The resulting magnetic field will slightly resemble the magnetic field from a bar magnet.

6 Ampere s Law #!!"!"! B ds = µ0 I We can use this to derive the B field for a straight wire:

7 Solenoid A solenoid is known as an electromagnet since it only behaves as a magnet when it carries current. " The field lines inside a solenoid are nearly parallel and close together. " This implies that the magnetic field inside a solenoid is strong. " The magnetic field outside the solenoid is weaker and nonuniform.

8 Solenoid The magne4c field lines from a solenoid strongly resemble those of a bar magnet. " The magnitude of the field inside a solenoid is constant at all points far from its ends. It can be given by: B = µo ni " where n is the number of turns per unit length. n = N /l

9 Torque on a Current Loop Let s say we have a rectangular loop of wire (of length b and width a) with a clockwise current immersed in a magne4c field. Fleft What would be the direction of the magnetic force on this wire loop? Use RHR1 on each section of the wire. Ftop I B field X X X X X X X X X X X X X X X Fright Fbot X X X X X For the top section, the force would point upward. For the bottom, the force would point downward. For the left, the force would point to the left. For the right, the force would point to the right.

10 Torque on a Current Loop The forces cancel out. So nothing will happen to the loop. " But, what if the wire loop was slightly tilted ( ) with respect to the magnetic field. " Now, we can create a torque due to these forces about a central axis. " The torque on the top section will be given by: " top =! r! F top sin# " top = a 2 F top sin#

11 Torque on a Current Loop The torque on the bovom sec4on will be given by: " bot =! r! F bot sin# " bot = a 2 F bot sin# " The force on the top and bottom wire will have the same magnitude given by: F top = F bot = IbBsin90 = IbB " Also, by right hand rule, the torques will point in the same direction, such that the torque from the top and bottom add.

12 Torque on a Current Loop " To find the total torque on the current loop, merely add the torques: " tot = " top + " bot " tot = a 2 F top sin# + a 2 F bot sin# " tot = af top sin# = a IbB ( )sin# " tot = IABsin# " where A is the area of the loop. " If we were to make N loops of this wire then our equation would become: " tot = NIABsin#

13 Magne4c Moment " The first part of right side of the last equation (NIA) is completely dependent on properties of the loop. " Because of this we define the magnetic moment vector,, of the wire: µ = NIA " The magnetic moment vector points perpendicular to the plane of the loop(s), a normal so to speak. " The angle will be between and B such that: " =! µ #! B =! µ! B sin$

14 Electric Motor " An application of this concept is the electric motor. " An electric motor converts electrical energy to mechanical energy (specifically rotational kinetic energy). " A simple electric motor consists of a rigid currentcarrying loop that rotates when placed in a magnetic field. " To provide continuous rotation in one direction, the current loop must periodically reverse, (AC current).

15 Torque A Current Loop Example A circular coil of wire with average radius 5.00cm and exactly 30 turns lies in a ver4cal plane. It carries 5.00A of current, counterclockwise when viewed from above. The coil is in a uniform magne4c field directed to the right, with a magnitude of 1.20T. Find the magne4c moment and the torque on the coil. Which way does the coil tend to rotate? B field " Answer " First, you must define a coordinate system. " Here we must define the normal for the coil, we say that it is perpendicular to the coil (out of the board).

16 " Answer Torque A Current Loop " Use the magnetic moment equation: µ = NIA " The area is for the circular loop giving us: ( ) µ = NI "r 2 µ = ( 30) ( 5A)" ( 0.05m) 2 =1.18 A# m 2 " Where the direction of this magnetic moment is the same as the direction of the normal of the coil. " For torque: " =! µ #! B =! µ! B sin$ " = ( 1.18 A# m 2 )( 1.20T)sin90 " =1.42 N # m

17 " Answer Torque A Current Loop " For direction, look at four separate points on the circular loop (top, left, bottom, and right parts): " Using RHR 1 on the top and bottom gives us: " Zero magnetic force (I and B are parallel in both instances). " Using RHR 1 on the left gives us: " A magnetic force that is out of board. " Using RHR 1 on the right gives us: " A magnetic force that is into the board. " So the coil will flip such that the right side goes into the board and the left side comes out of the board. B field

18 For Next Time (FNT) " Keep reading Chapter 22 " Con4nue working on the homework for Chapter 22