Ch 28 Magnetic fields

Ch 28 Magnetic fields - In full vector form the B field due...

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Unformatted text preview: In full vector form, the B field, due to a moving point charge, is given by: 2 ˆ 4 r r x v q B π μ = q v P φ Magnetic Fields It is of fundamental fact that a moving charge causes a magnetic field . Hence a current produces a magnetic field. The magnetic field around a current carrying wire can be measure by placing compasses around the wire. It is found that the magnetic field produces concentric circles around the wire having the same strength of magnetic B-fields . Now let us calculate the magnetic field do to a small segment d l of a current carrying conductor. 2 sin 4 r qv B φ π μ = The small current of wire can be treated like a moving point charge q, where the magnetic field from source to point P in space is given by : 2 sin 4 r qv B φ π μ = To get the total magnetic B field caused by the total wire, quite naturally the current elements must be added through superposition principle. Hence the small current element must correspond to a small B field based on the B from a single charge: 2 sin 4 r dqv dB φ π μ = A more appropriate way to write dq (the small charge) base on current, is: q (n A dl ). That is charge times a small volume times the density. 2 sin 4 r Adl nqv dB d φ π μ = The small magnetic field then becomes: But nqv d A is just the current in the element so: 2 sin 4 r Idl dB φ π μ = Where the angle φ is measure from the direction of current element dl to point P in space. The vector form of the magnetic field of a current element at point P simply becomes: 2 ˆ 4 r r x l Id B d π μ = The total magnetic field at point P is given by the integral: ∫ ∫ = = 2 ˆ 4 r r x l Id B d B π μ This is called the Biot Savart Law 2 ˆ 4 r r x l Id B d π μ = The field vectors d B and the magnetic field lines of a current element are exactly like B set up by a positive charge dq moving in the direction of the drift velocity. The field lines are circles in a plane perpendicular to d l and centered on the line of d l . Since we cannot have an isolated dl there is no way directly verify the above formula . However we can measure the total B field for a complete circuit. And the results are in agreement with the supposition . If matter is present near the wire, the field at point P will have an additional magnetic contribution due to the magnetization of the material. Let us find the magnetic field due to a long straight wire using the Biot- Savart law. The magnitude of B from a small segment of wire is given by: 3 4 r xIdy dB π μ = Substituting r 3 = (x 2 + y 2 ) 3/2 we have: ( 29 ∫ ∫ + = 2 / 3 2 2 4 y x xdy I dB π μ 2 ˆ 4 r r x l Id B d π μ = It is a fact that sin φ = sin (π – φ)....
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Ch 28 Magnetic fields - In full vector form the B field due...

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