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Rogers: Lectures based on Halliday, Resnick and Walker’sFundamentals of Physics, Copyright 2005 by Wiley and Sons Chapter 29 Ampere’s Law Sections 1-5 (exclude rail gun in section 3) 2 lectures So far in a magnetic field we have looked at -force on a moving charge or -force on current . Here we study how electric currents can create a magnetic field. We skip detailed form of relationship (Biot-Savart law - analogue of Coulomb’s law) and look at analogue of Gauss’ Law for electric fields. Ch29-1/22 Magnetic fields from currents Magnitude of field dB at point P due to a current length element ds is: where the permeability constant µo is given by The law of Biot-Savart is given by where r3 is really r2 to cancel r in numerator. Ch29-2/22Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 1
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Rogers: Lectures based on Halliday, Resnick and Walker’sFundamentals of Physics, Copyright 2005 by Wiley and Sons Magnetic fields from currents Book shows from Biot-Savart that the magnetic field around a long wire is given by Another RHR: with current element in right hand with thumb in direction of current => fingers are in direction of B Book also shows that at the centre of a segment of circular wire or for a full circle Ch29-3/22 Ampere’s Law Named after Andre Ampere (1775-1836) but actually advanced by James Maxwell Integral around a closed Amperian loop of dot product is related to enclosed current. Ch29-4/22Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 2
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Rogers: Lectures based on Halliday, Resnick and Walker’sFundamentals of Physics, Copyright 2005 by Wiley and Sons Ampere’s Law (cont) Can go around loop in arbitrary direction. In general, magnetic field from current in a single wire is in a plane | to wire (a consequence of the Biot-Savart law). In general do not know direction of B in the plane but can assign (usually in direction of integration) and if we get it wrong, sign will be negative. New Right Hand Rule Fingers along the direction of integration, the currents in the direction of right thumb are counted +ve and opposite the thumb are -ve as part of ienc Ch29-5/22 Ampere’s Law (cont) Note i3 not included since outside the loop => contribution to integral = 0 (which does not mean the contribution to B is zero - it isn’t). In this arbitrary case, we cannot solve for B, but we can in some special cases. Ch29-6/22Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 3
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Rogers: Lectures based on Halliday, Resnick and Walker’sFundamentals of Physics, Copyright 2005 by Wiley and Sons Magnetic field outside a long straight wire with current Wire carries i out of page. By symmetry, B is same everywhere at a given distance from wire, i.e. B is cylindrically symmetric. Use a concentric Amperian loop. Again, as consequence of Biot-Savart law, B is everywhere tangent to the loop. Generally we have: i.e. magnetic fields are tangent to circles around the wire and in a plane Ch29-7/22 Magnetic field outside a long straight wire with current Arbitrarily go counterclockwise and make use of alignments and constancy of B so that where rhr tells us enclosed current is +ve and B >0 => same direction as shown. Note magnetic field drops off as 1/r (as did electric field near a charged line conductor). Ch29-8/22Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 4
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Rogers: Lectures based on Halliday, Resnick and Walker’sFundamentals of Physics, Copyright 2005 by Wiley and Sons Magnetic field inside a long straight wire with current Consider magnetic field inside a long, straight current carrying wire. Assume current uniformly distributed. As before, directions of B and ds are as shown. So again: Enclosed current from simple ratio of areas rhr => +ve sign on enclosed current, so that i.e. inside the wire B builds linearly with r and is 0 at centre. Note that at r=R, both match Ch29-9/22 Question 4 Amperian loops with3 wires carrying current in directions as shown. Order the loops in terms of absolute magnitude of Which has largest magnitude? Note: this figure is different from one in book since it makes clear that loop b does Which has smallest magnitude? not include the middle wire. Ch29-10/22Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 5
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Rogers: Lectures based on Halliday, Resnick and Walker’sFundamentals of Physics, Copyright 2005 by Wiley and Sons Question 4 Amperian loops with3 wires carrying current in directions as shown. Order the loops in terms of absolute magnitude of Which has largest magnitude? Note: this figure is different from one in Ans: d is + 2i book since it makes clear that loop b does Which has smallest magnitude? not include the middle wire. Ans: b is 0 also a(+1) and c(-1) are equal in magnitude. Ch29-11/22 Sample problem Long conducting cylinder with inner radius a = 2.0 cm and outer b = 4.0 cm. Current out of page with current density J(r) = cr2 with c = 3.0E6 A/m2 where r in meters. What is magnetic field 3 cm from central axis? Key idea: symmetry => can use Ampere’s law. Draw Amperian loop shown in b at radius r = 3 cm. As before and enclosed current is Ch29-12/22Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 6
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Rogers: Lectures based on Halliday, Resnick and Walker’sFundamentals of Physics, Copyright 2005 by Wiley and Sons Sample problem From previous 2 results we have Note -ve sign comes because rhr tells us current out of screen is negative (thumb is into screen) -ve sign here means B is opposite to direction of integration (i.e. B is counterclockwise) Note result is independent of b and field is very weak given a current of 3.1 A Ch29-13/22 Force between two parallel currents Consider 2 long parallel wires carrying currents. Each creates a magnetic field and each experiences a force exerted by magnetic field of other. How big is this force? Magnitude of magnetic field caused by current ia at position of current ib is Curled-rhr tells us Ba is directed down. So what force does it produce? From before we have, for a length L of wire with current ib . Ch29-14/22Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 7
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Rogers: Lectures based on Halliday, Resnick and Walker’sFundamentals of Physics, Copyright 2005 by Wiley and Sons Force between two parallel currents RHR => it is directed towards other wire. Since L and B are at 90 degrees Do same thing for force on current ia caused by current ib Symmetry => it will be same. Two wires with parallel currents will attract each other. If the currents are anti-parallel, the forces will be repulsive. Ch29-15/22 Force between two parallel currents The formal definition of ampere, one of 7 SI base units. The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross section and placed 1 m apart in vacuum would produce on each of these conductors a force of magnitude 2x10-7 newton/m of wire length. Ch29-16/22Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 8
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Rogers: Lectures based on Halliday, Resnick and Walker’sFundamentals of Physics, Copyright 2005 by Wiley and Sons Magnetic field of a solenoid A solenoid is a long, tightly wound helical coil of wire. We assume length >> diameter. Diagram at right is a stretched out solenoid. Close to each turn, the wire acts magnetically like a long straight wire => magnetic field circular around it dropping off as 1/r. Tends to cancel between turns and lead to parallel lines of force between. Ch29-17/22 Magnetic field of a solenoid (cont) current out of screen at top, in at bottom. For an ideal solenoid (touching square wires of infinite length) we get a uniform field in interior, parallel to axis (not a trivial observation). At points like P outside the solenoid, the field from the upper wires tends to cancel that from the lower wires, and for a perfect solenoid they do so exactly => 0 field outside the solenoid. This is a good assumption away from the ends for length >> diameter. Direction of field in interior follows RHR where fingers point in direction of current. Ch29-18/22Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 9
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Rogers: Lectures based on Halliday, Resnick and Walker’sFundamentals of Physics, Copyright 2005 by Wiley and Sons Magnetic field of a solenoid (cont) current out of screen at top, in at bottom. Above is a realistic solenoid. Field is stronger and close to parallel inside and much weaker outside. In practice, solenoids can be used to set up a uniform field over a fairly large volume, just like a parallel plate capacitor sets up a known and uniform electric field. Ch29-19/22 Magnetic field of a solenoid (cont) current out of screen at top, in at bottom. Ideal solenoid: B constant and parallel axis in the interior and 0 outside. Consider the Amperian loop shown. Let n be turns/unit length This eqn actually holds reasonably well for real solenoids. Note that B is independent of length or diameter. Ch29-20/22Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 10
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Rogers: Lectures based on Halliday, Resnick and Walker’sFundamentals of Physics, Copyright 2005 by Wiley and Sons Sample problem Solenoid length 1.23 m, inner diameter 3.55 cm and current of 5.57 A. It has 5 layers of turns, each with 850 turns in the length. What is B in the centre of the solenoid? Key idea: The magnetic field does not depend on the diameter of the solenoid, and therefore we can treat each of the 5 layers as the same solenoid. Hence Ch29-21/22 Magnetic field of a toroid A toroid is a hollow solenoid bent into a circle. What is B inside the `hollow bracelet’? From symmetry lines of B form concentric circles inside the hollow. Amperian loop at radius r. Total of N turns. Ampere’s law => No longer constant but still independent of size of toroid. Direction defined y the RHR when fingers follow direction of current. current out of page B= 0 anywhere outside the toroid since current outside, in on inside flowing thru any plane is 0. Ch29-22/22Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 11
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