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Section8_Biot's_Law_soln.notebook
May 01, 2012
Section 8: Biot's Law
The Magnetic Field (B) around a Straight Current Carrying Conductor
In this lesson you will • state Biot's Law in word and mathematical form • apply Biot's Law in numerical exercises that deal with straight current carrying conductors 1
Section8_Biot's_Law_soln.notebook
May 01, 2012
Ampere's Law
Ampere's Law can best be done with a calculus treatment. We're will just scratch the surface here. Recall Hans Christian Oersted (1777­1851) discovery of a magnetic field around a straight conductor.
Notice:
larger the current ⇒denser the lines⇒ the further they reach.
larger the current ⇒the stronger the field, or B α I. Density of the lines decrease as the distance from the conductor increases. The greater the distance from the conductor ⇒the weaker the field. This is an inverse relationship, written as B α 1/r
Combining the two proportions gives By inserting a constant of proportionality, the following equation can be written:
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Section8_Biot's_Law_soln.notebook
May 01, 2012
André Marie Ampere (1775­1836) was a French physicist and philosopher. Ampere was interested in extending Oersted's discovery to a more general situation where the conductor is not necessarily a straight one, and the path around the conductor is not a nice circle. The latter of these two conditions is shown in the next picture.
In the second part of the picture, the path is broken into tiny segments, ΔL, and each tiny path segment has a small parallel component of the magnetic field strength B associated with it.
Ampere was able to show that the sum of the products of was directly proportional to the current I flowing through the closed path. That is , In other words, the larger the current, the bigger would be the sum of the little products. The proportionality can be written as an equation when a constant of proportionality is inserted. Ampere was able to show that the constant was a very special one: it was μ !! Recall that μ is the magnetic permeability of the substance in which the field is located. The mathematical form of Ampere's Law is therefore
Now for the moment of truth: we will apply Ampere's general statement to Oersted's special case where the pattern of the magnetic field is a perfect circle around a straight conductor.
From Ampere’s Law
ΣB║ΔL = μI
But, B is constant if r is constant. Therefore, B(ΣΔL) = μI
But, ΔL, for a circle = 2πr
Therefore,
B(2πr) = μI
B = μI/2πr or, B = (μ/2π) (I/r)
Note this corresponds with Oersteds eqn!
[ B = k I/r, where k = μ/2π]
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Section8_Biot's_Law_soln.notebook
May 01, 2012
Biot’s Law (attributed to Jean­Baptiste Biot (1774­1862))
B = μI/2πr
The 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, with the constant of proportionality being μ/2π. If the field is in free space, the μ is written as μo where μo = 4π x 10­7 T∙m/A. Air can be considered to be free space. The units of μ are T∙m/A where T = tesla, m = metre and A = ampere.
Examples:
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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.
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Section8_Biot's_Law_soln.notebook
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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 10­5 T. What is the resistance of the conductor in ohms?
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Two parallel wires each carry 5.0 A of current in opposite directions. What is the magnetic field strength midway between the wires if the wires are 10 cm apart?
May 01, 2012
Step 1: Draw a picture.
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Section8_Biot's_Law_soln.notebook
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May 01, 2012
Repeat practice exercise #3 with the current in the wires running in the same direction.
A closing note of caution
Ampere's Law and Biot's Law deal with magnetic field strength (B). Conspicuously absent from Ampere's Law and Biot's law is any mention of magnetic force (F). As soon as you read an exercise in which you are given the magnetic force (F), or are asked to calculate F, you will have to apply some other formula. You already know F = BIL sinθ section 7 and you will meet one other.
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Section8_Biot's_Law_soln.notebook
May 01, 2012
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Section8_Biot's_Law_soln.notebook
May 01, 2012
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