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Electricity & Magnetism
Seb Oliver Lecture 15: Biot-Savart Law
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Biot-Savart Law
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Introduction We have discussed how an existing magnetic field influences moving charges (and thus currents) We have not yet discussed the origin of magnetic fields We will now see that currents (moving charges) produce magnetic fields This can be thought of as the basic mechanism by which all magnetic fields are produced
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History 1819 Hans Christian Oersted discovered that a compass needle was deflected by a current carrying wire Then in 1920s Jean-Baptiste Biot and Felix Savart performed experiements to determine the force exerted on a compass by a current carrying wire There results were as follows …
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Jean-Baptiste Biot & Felix Savart’s Results
dB the magnetic field produced by a small section of wire ds a vector the length of the small section of wire in the direction of the current r the positional vector from the section of wire to where the magnetic field is measured I the current in the wire angle between ds & r dB r ds
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Biot & Savart’s Results
dB perpendicular to ds dB perpendicular to r |dB| inversely proportional to |r|2 |dB| proportional to current I |dB| proportional to |ds| |dB| proportional to sin q
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Biot – Savart Law All these results could be summarised by one “Law”
Putting in the constant Where m0 is the permeablity of free space
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Magnetic Field from Biot-Savart Law
We can use the Biot-Savart law to calculate the magnetic field due to any current carrying wire dB2 dB1 dBi dsi r2 ri B = dB1+dB2+…+dBi I.e. B =SdB r1 ds2 ds1
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One Example of using the Biot-Savart Law
Direction of the field around a long wire
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Magnetic Field from Biot-Savart Law
We can use the Biot-Savart law to see the direction of the field due to a wire segment dB1 dB1 r1 r1 ds1 ds r dB
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Another Right-Hand Rule
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Magnetic Field from Biot-Savart Law
dB1 r1 c.f. Of course there is no such thing as an isolated current segment!
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Using the Biot-Savart Law
Example 1: A wire loop
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Examples of using the Biot-Savart Law
Some examples are quite hard and show how difficult it is to use this Law In a real situation you might use a computer to calculate the magnetic field due to the wire Next week we will discover Amperes Law which is often much easier to use
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Magnetic Field from a Current Loop
ds Direction: r Magnitude: B is always out of screen ds is always to r ds r dB
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Using the Biot-Savart Law
Example 2: A tight coil
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Magnetic Field from a tight coil I.e. many current loop
ds Direction: r Magnitude: as for a single loop 1 loopCurrent I N loops Current NI
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Using the Biot-Savart Law
Example 3: A long Straight wire
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Magnetic Field from a Straight wire
q ds x
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Magnetic Field from a Straight wire
Magnitude: Direction: B is always out of screen It turns out it is easier if we get everything in terms of q ds r dB
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Magnetic Field from a Straight wire
Magnitude: … Now we have to use calculus!
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Other examples of Magnetic field
Centre of a wire loop radius R Centre of a tight Wire Coil with N turns Distance a from long straight wire
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Magnetic Force Between Two Parallel Wires
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Magnetic Force Between Two Parallel Wires
A current carrying wire in a magnetic field feels a magnetic force A current carrying wire generates a magnetic field Thus two current carrying wires will exert a force on each other
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Magnetic Force Between Two Parallel Wires
Magnetic field at wire 1 from wire 2 I2 B2 Magnetic Force on wire 1 due to B2 (everything )
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Magnetic Force Between Two Parallel Wires
Force on wire 1 due to wire 2 From Newton’s 3rd Law (& symmetry) Force on wire 2 due to wire 1 Force / unit length
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Definition of Ampere If the magnitude of the force per unit length between 2 parallel wires carrying identical currents and separated by 1m is 2×10-7 N/m then the current in each wire is 1A Definition of a Coulomb: If a current of 1A is passing through a wire then 1C of charge passes a surface in 1s
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Quiz I1 = 2A, I2 = 6A (a) F1 = 3F2 (b) F1 = F2 (c) F1 = F2/3
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Summary Biot-Savart Law Force between two wires Definition of Ampere
(Field produced by wires) Centre of a wire loop radius R Centre of a tight Wire Coil with N turns Distance a from long straight wire Force between two wires Definition of Ampere
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