1. © 2001-2005 Shannon W. Helzer. All Rights Reserved. 1 Chapter 29 – Magnetic Fields Due to Current

2. © 2001-2005 Shannon W. Helzer. All Rights Reserved. 2 Biot Savart Law WS 11 #1  Whenever a current flows through a wire, a magnetic field is induced about the wire.  The direction of this magnetic field may be found by applying the right hand rule to the cross product below.  Point your thumb in the direction the current is flowing.  The magnetic field wraps around the wire in the same direction as your fingers wrap around the wire.  When the current is going up (remember, current is in the opposite direction as the electrons are moving), we find that the magnetic field around the wire is counterclockwise.  When the current is going down, we find that the magnetic field around the wire is clockwise.  We will determine the magnitude of the magnetic field on the next slide.

3. © 2001-2005 Shannon W. Helzer. All Rights Reserved. 3 Biot Savart Law – Magnetic Field of a long, Straight Wire WS 11 #4  In this problem we will determine the magnetic field generated at point P by the upper half of a long current carrying wire.  First determine the magnetic field direction.  The vector ds is in the same direction as the current.

4. © 2001-2005 Shannon W. Helzer. All Rights Reserved. 4 Biot Savart Law – Magnetic Field of a long, Straight Wire WS 11 #4

5. © 2001-2005 Shannon W. Helzer. All Rights Reserved. 5 Biot Savart Law – Magnetic Field of a long, Straight Wire WS 11 #4  We have just found the magnetic field generated by the upper half of the wire.  In order to determine the magnetic field generated by the entire length of the wire at point P, simply multiply our result by two.

6. © 2001-2005 Shannon W. Helzer. All Rights Reserved. 6 Magnetic Field due to a Current Carrying Arc WS 11 #5  Determine the magnetic field at point P due to an arc shaped current carrying wire.  Remember,  must be in radians.

7. © 2001-2005 Shannon W. Helzer. All Rights Reserved. 7 Magnetic Field Example WS 11 #6  Determine the magnetic field at point P due to current carrying wire shown.  The arc in the wire covers 3  /4 radians.  This problem may be done in three segments.  What is the angle between R 1 and ds 1 ?  What is the angle between R 3 and ds 3 ?  We already derived the equation for the arc portion of the wire.

8. © 2001-2005 Shannon W. Helzer. All Rights Reserved. 8 Example Problem WS 11 #7  Two long, parallel wires separated by a distance of 6.0 cm carry current as shown.  Both wires carry a current of 17.5 A.  Point P is located 1/3 the distance from the left wire and 3.0 cm above the plane through both wires.  What is the magnetic field (magnitude and direction) at point P?  We begin by using the equations we derived for the long straight wires in order to calculate the magnitudes of the magnetic fields.  We have already determined the directions of the magnetic fields at point P.

9. © 2001-2005 Shannon W. Helzer. All Rights Reserved. 9 Example Problem WS 11 #7

10. © 2001-2005 Shannon W. Helzer. All Rights Reserved. 10 Example Problem WS 11 #7 VectorMagnitudeAnglex componenty componentQuad --------------

11. © 2001-2005 Shannon W. Helzer. All Rights Reserved. 11  Two wires carry currents as shown below.  Calculate the force of attraction or repulsion that exists between these two wires.  First let us determine the direction of the magnetic field around these two wires by using our right hand rule.  Recall that the equation used to find the force on a charge carrying conductor is as follows.  Lets consider the force on wire 2 due to wire 1.  The magnetic field generated by wire 1 causes the force experienced by wire 2.  Note the direction of this magnetic field acting on wire 2.  Applying the cross product, we observe that the force acting on wire 1 is directed towards wire 2.  Likewise, the force on wire 2 is directed towards wire 1. Force Between two Wires WS 11 #8 & 9

12. © 2001-2005 Shannon W. Helzer. All Rights Reserved. 12  What do you suppose would happen if the current in one of the wires changes direction?  Applying the cross product below and the right hand rule, we find that the force generated by the magnetic fields cause the wires to move away from each other.  In summary, if the current in both wires moves in the same direction, then the wires will attract each other.  If the current in the wires moves opposite direction, then the wires will repel each other. Force Between two Wires WS 11 #10 (Homework)

13. © 2001-2005 Shannon W. Helzer. All Rights Reserved. 13 WS 11 #12 (Homework)  The figure to the right has an outer radius r 1 = 16.50 cm and an inner radius r 2 = 8.85 cm. The arc subtends an angle of 98.0 .  The current flowing through the object is I = 0.523 A.  Determine the magnetic field (magnitude and direction) acting at point P.

14. © 2001-2005 Shannon W. Helzer. All Rights Reserved. 14 WS 11 #13  The figure below shows different arrangements of four cross-sectional views of four current carrying wires.  Which, if any, of these has a net zero magnetic field at the center?  Explain your reasoning.

15. © 2001-2005 Shannon W. Helzer. All Rights Reserved. 15 WS 11 #14 (Homework)  The figure to the right shows a current carrying conductor.  This conductor splits into a circular and square segment shown.  The current I = 3.80 A before it splits.  The circle has a radius of 2.54 cm and the square has sides equal to 2.54 cm.  What is the magnetic field (magnitude and direction) at the center of these two shapes?

16. © 2001-2005 Shannon W. Helzer. All Rights Reserved. 16  Ampere’s law is used to determine the net magnetic field produced by any distribution of current carrying conductors.  Consider the three cross sections of current carrying wires below.  We use an Amperian Loop in order to determine the net magnetic field generated by the enclosed wires.  Ampere’s Law in equation form is as follows where i enc is the net current enclosed by the Amperian loop.  The direction of the arrows on the Amperian loop are randomly chosen.  Likewise, the vector B and angle  are randomly chosen.  In order to determine the magnitude of the currents in the wires enclosed b the Amperian loop, we will use another right hand rule.  Place your hand in a position where your fingers point in the same direction as the arrows on the loop.  If the current in the wire points in the same direction as your thumb, then the current in that wire is considered to be positive. Ampere’s Law WS 11 #15 & 16

17. © 2001-2005 Shannon W. Helzer. All Rights Reserved. 17  As a result, i 1 would be positive and i 3 would be negative.  Just as we did with Gauss’ law, we use symmetrical bodies as our Amperian surfaces. Ampere’s Law WS 11 #15 & 16

18. © 2001-2005 Shannon W. Helzer. All Rights Reserved. 18  Find the magnetic field produced by the current carrying wire of radius r shown below.  What is the angle between ds and B?  Do not forget the right hand rule for the Amperian surface. Ampere’s Law – Long Straight Wire WS 11 #17

19. © 2001-2005 Shannon W. Helzer. All Rights Reserved. 19  Whereas an electric field cannot exist inside a current carrying conductor, a magnetic field can.  Find the magnetic field a distance of r from the center of a current carrying conductor of radius R (r < R).  Only a fraction of the total current passes through our Amperian surface Ampere’s Law – Inside a Long Straight Wire WS 11 #18

20. © 2001-2005 Shannon W. Helzer. All Rights Reserved. 20 Other Magnetic Fields  We have already seen the magnetic field generated by a current carrying wire.  In the next two slides we will see the magnetic fields generated by different geometries of current carrying conductors.

21. © 2001-2005 Shannon W. Helzer. All Rights Reserved. 21 Other Magnetic Fields  A coiled wire is known as a solenoid.  The magnetic field generated by a solenoid looks like the one appearing below.

22. © 2001-2005 Shannon W. Helzer. All Rights Reserved. 22 Other Magnetic Fields

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