Other Formulae for Electromagnetism. Biot's Law Force on moving charges

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1 Other Formulae for Electromagnetism Biot's Law Force on moving charges 1

2 Biot's Law. Biot's Law states that the magnetic field strength (B) is directly proportional to the current in a straight conductor, and inversely proportional to the perpendicular distance (r) away from the conductor. B I r 2

3 The constant of proportionality being µ/(2p), where the µ is the magnetic permeability of the substance in which the field is located. NOTE: If the field is in free space, the µ is written as µ o where µ o = 4p x 10-7 T m/a (where T = tesla, m = metre and A = ampere) Air can be considered to be free space. 3

4 Mathematically: B I r 2 p This law is attributed to Jean- Baptiste Biot ( ). 4

5 Practice 1. Find the magnetic field strength (B) in air 7.0 mm away from a straight conductor in which there is a current of 2.0 A 7Tm 4p10 o B I A 2.0A 2 p 3 r 2 p ( m) B T 5

6 2.When a potential difference of 12.0 V is applied to a straight conductor, the magnetic field strength (B) 2.0 cm from the conductor is 3.0 x 10-5 T. What is the resistance of the conductor in ohms? o 2prB B I I 2 p r o 5 2 p(0.02 m)( T ) I 7Tm 4p10 A 3A Answer: The resistance of the conductor is 4.0 W. 6

7 3.Two parallel wires each carry 5.0 A of current in opposite directions. A) Which way are the wires forced to move? Answer: The wires will move apart because there is a stronger magnetic field in between the wires. 7

8 3.Two parallel wires each carry 5.0 A of current in opposite directions. B) What is the magnetic field strength midway between the wires if the wires are 10 cm apart? B 1 o I 2 p r 8

9 4. Repeat practice exercise #3 with the current in the wires running in the same direction. Solution: A) The wires will move together. B) At mid-distance between the wires, B l and B 2 cancel each other so that the net magnetic field strength is 0.0 T. 9

10 5. Two 1m parallel wires each carry 1.0 A of current in opposite directions. What is the force that acts on one of the wires?

11 6. How far from a conductor carrying 5.0 A of current is a second wire with a current of 10.5 A if the force between the two wires is 2 x 10-5 N/m

12 Magnetic Force on Moving Charges Thus far, we have been saying that the force on a current-carrying conductor in an external magnetic field is F = BIL sinq. The key phrase here is current-carrying. If there were no current, there would be no force on the conductor So, the force is really exerted on the current. The conductor provides a path for the current. 12

13 If a stream of electrons were shot through a magnetic field the force on the stream would still be F = BIL sinq, where, in this case, L would be the length of the stream that falls within the magnetic field. To simplify things, instead of picturing a current of electrons passing through a magnetic field, imagine a single electron, q, being fired into the field. Of course, the single charge, q, doesn't have to be an electron. It could be a proton, or ion. 13

14 If q is traveling at a speed v and takes a time of t to travel a distance L, what is the relationship between L, v and t? Since distance = speed x time, L = vt. Think back for a moment to a stream of electrons. If there are n charges of size q in the stream, what is the total charge (Q) in the stream? The total charge will be Q = nq. (Recall Q = ne) 14

15 Think back even further to the expression for current, I. How did we write it? Q I t Write an expression for I by using nq and t. nq I t Use this expression above and L = vt to re-write F = BIL sin q. nq F B vt )sin q t Bnqv sin q 15

16 The previous expression F Bnqv sin q gives the force on n charges. But we are looking for the force on a single charge, that is, n=1. The expression becomes F Bqv sin q where B is the magnetic field strength in tesla (T), q is the magnitude of the charge in coulombs (C) that is moving at a velocity v in m/s, and q is the angle between v and B 16

17 Direction of the Force To determine the direction of the force on a negative charge that is passing through a magnetic field, just apply left-hand rule #3: Point your fingers in the direction of the magnetic field, your thumb in the direction that the charge is moving, your palm points in the direction of the force on the charge. 17

18 Direction of the Force While the left hand rule will work to show the deflection of a stream of electrons, the right hand rule will be useful for a stream of positive particles. 18

19 Example: 1. A magnetic field of 44.0 T is directed into a computer screen. A particle with a negative charge of 2.0 x C is shot into the field from the right, making an angle of 90 o with the field lines. If the particle is moving at 5.4 x 10 7 m/s, what magnetic force does it experience? Solution: F= 4.8 x 10-9 N Directed upwards 19

20 p do #6--#8. p do #12, #14 p do #31 20

21 More fun stuff! If a charge is shot in horizontally (or perpendicular to the lines of force), it will follow a circular path. This is because the moving charge has a circular field concentric around its direction of motion. This circular field and the permanent field reinforce each other on one side of the charge, therefore forcing it into circle. 21

22 N - Pole N - Pole e e S - Pole S - Pole Electron shot in perpendicular to magnetic field Electron shot in at angle to magnetic field 22

23 What would happen if the charged particle was a proton? N - Pole N - Pole S - Pole S - Pole 23

24 Describe the behaviour of beams of charged particles passing through a magnetic field. N - Pole e S - Pole This is how a TV picture tube works. 24

25 Centripetal Magnetic Force For a particle moving at a constant speed and experiencing a constant magnetic force at 90 o to its motion traces a circular path. What is providing the centripetal force? The magnetic force is. 25

26 Example: An electron is shot perpendicularly into a magnetic field of strength T with a velocity of m/s. What is the radius of the electron s path inside the magnetic field? F net Fcentripetal 2 mv r mv Bqv 2 sin F magnetic F Bqv q magnetic sin r q r Bq S ince q r mv sin q mv Bq 90 o 26

27 Example: An electron is shot perpendicularly into a magnetic field of strength T with a velocity of m/s. What is the radius of the electron s path inside the magnetic field? r mv Bq 31 6 ( kg)( m / s ) 5 ) T C ) 27

28 Review example 8 page 654 A) What is the velocity of an alpha particle moving in a circular path of radius 10.0 cm in a plane perpendicular to a 1.7 T magnetic field? Solution: An alpha particle is a particle consisting of two neutrons and two protons that is identical to the helium nucleus and is emitted during certain radioactive transformations. What is the charge on an alpha particle? q C 2 ( ) C 19 28

29 m kg r 10.0cm 0.100m B 1.7T mv r mv rbq Bq v v rbq m 19 m T C v kg m s 29

30 B) If this alpha particle is accelerated by the application of an electric field over a set of parallel plates, what voltage is required to accelerate the alpha particle from rest? Recall: V E q and Kinetic Energy is given by 2 mv V 2q E 1 mv 2 2 V kg C 27 6 m s V 30

31 Page 665 #28,29,30 31

Section 9: Magnetic Forces on Moving Charges

Section 9: Magnetic Forces on Moving Charges In this lesson you will derive an expression for the magnetic force caused by a current carrying conductor on another current carrying conductor apply F = BIL