A Guide to Biot-Savart Law
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Probably one of the hardest, and most confusing, of the four electromagnetic equations is the Biot-Savart Law (pronounced bee-yo-suh-var).
This law is easily seen as the magnetic equivalent of Coulomb's Law. What it basically states is that the magnetic field decreases with the square of the distance from a "point of current" or current segment. Where it differs is the fact that a point of current is much harder to achieve than a point charge.
As mentioned earlier, the Biot-Savart law deals with a current element. A
is like a magnetic element in that it is the current multiplied by distance. However a current element, by its very definition, cannot exist in a single point. Therefore, we must take the derivative of the current element and integrate a path of point-current elements. Stay with me, this becomes less confusing as it goes on.
Initially, let's try to derive the Biot-Savart Law from its similarity to Coulomb's Law and other facts that we already know. First, we'll start with an expression for an electric field around a point charge based on Coulomb's Law:
If we exchange
(I is always constant in a wire) and dl makes it a point-current element or current segment
(infinitesimals must be conserved),
then we get the very basics of the Biot-Savart Law.
Our next step will be to decide what expression will replace "
in Coulomb's Law is
is always on the opposite side of the fraction with
on these laws, the "
" for Biot-Savart law should be
So we now have
One final consideration that we must consider is that the current element has something that a point charge doesn’t have -- a direction. Since a magnetic field is strongest when it is at right angles to the current, we have to include the cross product of the direction of the radius,
is the angle between
That wasn’t so hard, was it? You might want to take a breather before continuing. Rested? Then let's use the Biot-Savart Law to find the magnetic field around a current carrying wire and at the center of a current loop.
Magnetic Field Around a Current Carrying Wire
First we are going to find the magnetic field at a distance R from a long, straight wire carrying a current of I. To do this, we must determine the proper use of Biot-Savart.
Pulling out all of the terms that aren’t related to distance will give us
This wire is long, so we are going to pretend that it is infinite in length.
Using symmetry principles, we are going to cut our wire in half and change our limits. Later, these symmetry properties will allow us to double our final B-field's value.
is the distance from the point-current element to the closest point of the wire to the point, and
is the distance from the point to the wire, and
is the distance from the point-current element to the point.
Now we use the trigonometric identity
to replace r
Now we need to replace our differential
. For this we use the trigonometric identity:
Now we substitute and integrate; but, because our differential has changed, so our limits must change. When
becomes infinitely small and approaches zero; when
You should now recognize this result from our previous lesson on
Magnetic Field at the Center of a Current-Carrying Loop
Let’s try something else. What would be the magnetic field at the center of a current carrying loop? Let us assume that the wire is a loop with a radius R and carries a current of
image courtesy of John Hopkins University
Physics Lecture Demonstrations
is always perpendicular to the direction of the current, we do not need to worry about messy integration.
Furthermore, since we are in a circular loop,
is equal to
. So we end up with
It is easily possible to find the magnetic field in many other geometries. The Biot-Savart Law is much, much, much more accurate than Ampere's Law (as its applications involve fewer assumptions). However, it is also much harder to apply. Therefore, it will tend to be the law used when Ampere's Law doesn't fit. For
, find other geometries of wires to practice with because nobody likes Biot-Savart.
Details for the derivation of a magnetic field around a current carrying wire were provided courtesy of Carlos Valera, May 2011.
Forces Between Ceramic Magnets
Magnetic Field in a Solenoid
Mass of an Electron
RC Time Constants
A Comparison of RC and RL Circuits
A Special Case of Induction
Dielectrics: Beyond the Fundamentals
Eddy Currents plus a Lab Simulation
Electric Field Strength vs Electric Potential
Electricity and Magnetism Background
Famous Experiments: Cathode Rays
Introduction to Magnetism
Magnetic Field Along the Axis of a Current Loop
Magnetic Forces on Particles (Part II)
Magnetism: Current-Carrying Wires
Meters: Current-Carrying Coils
Spherical, Parallel Plate, and Cylindrical Capacitors
Torque on a Current-Carrying Loop
The Tree House
Meters and Motors
Magnetic Forces on Current-Carrying Wires
Magnetic Forces on Moving Charges
Practice with Ampere's Law
36A: Magnets, Magnetic Fields, Particles
36B: Current Carrying Wires
Electric Field Strength vs Electric Potential
Exercises on Current Carrying Wires
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Lawrence F. Camarota
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