Chapter 28 Source of Magnetic Field

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1 Chapter 28 Source of Magnetic Field Lecture by Dr. Hebin Li

2 Goals of Chapter 28 To determine the magnetic field produced by a moving charge To study the magnetic field of an element of a current-carrying conductor To calculate the magnetic field of a long, straight, currentcarrying conductor To study the magnetic force between current-carrying wires To determine the magnetic field of a circular loop

3 The magnetic field of a moving charge A moving charge generates a magnetic field that depends on the velocity of the charge. The magnetic field magnitude at P is B = μ 0 q v sin φ 4π r 2 The direction is given by the right-hand rule. In the vector form, the magnetic field due to a moving point charge is Where the constant μ 0 = 4π 10 7 T m/a

4 Magnetic force between moving protons Find the magnetic force between two moving protons. The magnetic field is The magnetic force is

5 Example: An electron and a proton are each moving at 845 km/s in perpendicular paths as shown in the figure. At the instant when they are at the positions shown in the figure, find the magnitude and direction of (a)the total magnetic field they produce at the origin; (b)the magnetic field the electron produces at the location of the proton; (c)the magnetic force that the electron exerts on the proton.

6 Magnetic field of a current element The total magnetic field of several moving charges is the vector sum of each field. For a short segment d l of a currentcarrying conductor, the total moving charge dq in the segment is dq = nqadl The magnitude of the field at field point P is db = μ 0 dq v d sin φ 4π r 2 = μ 0 n q v d Adl sin φ 4π r 2 Here I = n q v d A, so db = μ 0 4π Idl sin φ r 2 The law of Biot and Savart:

7 Example: Find the magnetic fields at P 1 and P 2. At P 1 At P 2

8 Example: Two parallel wires are 5.00 cm apart and carry currents in opposite directions, as shown in the figure below. Find the magnitude and direction of the magnetic field at point P due to two 1.50-mm segments of wire that are opposite each other and each 8.00 cm from P.

9 Magnetic field of a straight current-carrying conductor If we apply the law of Biot and Savart to a long straight conductor, at point P, the field caused by each element of the conductor points into the plane of the page. The total field is the sum of fields caused by all elements.

10 Magnetic field of a straight current-carrying conductor If we apply the law of Biot and Savart to a long straight conductor, at point P, the field caused by each element of the conductor points into the plane of the page. The total field is the sum of fields caused by all elements. r = x 2 + y 2 ; sin φ = x/ x 2 + y 2 db = μ 0I 4π B = μ 0I 4π B = μ 0I 4π xdy (x 2 + y 2 ) 3/2 a a xdy (x 2 + y 2 ) 3/2 2a x x 2 + a 2 When a, B = μ 0I 2πx

11 Magnetic fields of long wires Investigate the magnetic field produced by the two wires at points P 1, P 2 and P 3.

12 Example: Four very long, parallel power lines each carry 100-A currents. A cross-sectional diagram of these lines is a square, 20.0 cm on each side. For each of the three cases shown in the figure below, calculate the magnetic field at the center of the square.

13 Force between parallel conductors The conductors attract each other if the currents are in the same direction and repel if they are in opposite directions. The field produced by one conductor at the position of the other one is B = μ 0 I 2πr. The magnitude of the force is F = I LB = μ 0II L 2πr So the force per unit length F/L is

14 Example: Two long, parallel wires hang by 4.00-cm-long cords from a common axis. The wires have a mass per unite length of kg/m and carry the same current in opposite directions. What is the current in each wire if the cords hang at an angle of 6.0 with the vertical?

15 Magnetic field of a circular current loop The Biot Savart law gives The x and y components are The total field has only x component

16 Magnetic field of a coil The direction of the magnetic field produced on the axis of a currentcarrying coil can be determined by the right-hand rule. For a coil consisting of N loops, the total field is N times the field of a single loop: At the center of N circular loops

17 Defining the Ampere The force between two straight, parallel, current-carrying conductors is the basis of the official SI definition of the ampere:

18 Ampere s law (general statement) The line integral of a magnetic field along a closed path is given by the Ampere s law

19 Application of Ampere s law The magnetic field due to a long, straight, current-carrying conductor.

20 Application of Ampere s law A long solenoid has n turns per unit length and carriers current I. Use Ampere s law to find the magnetic field at or near the center of the solenoid.

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