The Biot-Savart Law: Definition & Examples

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  • 0:01 What Is the Biot-Savart Law?
  • 1:45 Simplified Equation
  • 3:25 Example Calculation
  • 4:58 Lesson Summary
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Lesson Transcript
David Wood

David has taught Honors Physics, AP Physics, IB Physics and general science courses. He has a Masters in Education, and a Bachelors in Physics.

Expert Contributor
Elaine Chan

Dr. Chan has a Ph.D. from the U. of California, Berkeley. She has done research and teaching in mathematics and physical sciences.

After watching this lesson you will be able to explain what the Biot-Savart Law is, and use the simplified version to solve problems. A short quiz will follow.

What Is the Biot-Savart Law?

Electric fields and magnetic fields might seem different, but they're actually part of one larger force called the electromagnetic force. Charges that aren't moving produce electric fields. But when those charges do move, they also create magnetic fields. For example, a magnet is only a magnet because of moving charges inside it. And charges moving in an electric wire also produce magnetic fields. If you move a compass near to an electric wire, you'll find that the compass needle changes direction.

The Biot-Savart Law is an equation that describes the magnetic field created by a current-carrying wire, and allows you to calculate its strength at various points.

To derive this law, we first take this equation for the electric field. This is the full version, where we use muu-zero over 4pi instead of the electrostatic constant k. Since we're looking at a wire, we replace the charge q with I dl, which is the current in the wire, multiplied by a length element in the wire. Basically, it's treating this little chunk of the wire as our charge. And we also replace the electric field E with a magnetic field element dB because a moving charge produces a magnetic field, not an electric field.

Last of all, we have to realize that a current has a direction (unlike a charge). So we need to make sure the direction of the current affects our result. We do that by adding sine of the angle between the current and the radius. That way, if the wire is curvy, we'll take that into account. And that's it - that's the Biot-Savart law.


Simplified Equation

Using the Biot-Savart Law requires calculus. That's why there's a dB and dl. Those are infinitesimal magnetic field elements and wire elements. So we'd have to integrate with respect to those elements. But we can use a simpler version of the law for a perfectly straight wire.

If we straighten out the wire and do some calculus, the law comes out as muu-zero I divided by 2pir. Or in other words, the magnetic field, B, measured in teslas is equal to the permeability of free space, muu-zero, which is always 1.26 x 10^-6, multiplied by the current going through the wire, I, measured in amps, divided by 2pi times the radius away from the wire, r, measured in meters. So this equation helps us figure out the magnetic field at a radius r from a straight wire carrying a current I.


The equation gives us the magnitude of the magnetic field, but a magnetic field is a vector, so what about the direction? The magnetic field created by a current-carrying wire takes the form of concentric circles. But we have to be able to figure out if those circles point clockwise or counter-clockwise (say, from above). To do that we use a right-hand rule.

I want you to give the screen a thumbs up, right now. I'm serious - give the screen a thumbs up with your right hand. It has to be with your right hand. If you point your thumb in the direction of the current for this wire, your fingers will curl in the direction of the magnetic field. They'll follow the arrows of the concentric circles. And that's how you figure out the direction.


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Additional Activities

Using the Biot-Savart Law (Examples)


The Biot-Savart law is an integral that calculates the vector magnetic field B at a point in space due to an electric current I flowing in a conductor. The vector r from the origin of the vector current element to the point in space has magnitude r and makes an angle \theta to the vector current element dl. The direction of the magnetic field B is determined by the right hand (screw) rule: Let fingers of the right-hand point in direction of the current element dl and turn fingers to meet the r vector, then your thumb is in the direction of B.

Let's use Biot-Savart to find B at the center of a circular loop of radius R. The sine of the angle everywhere is one because the position vector and current element are perpendicular to each other, so we only need to integrate around the circumference of the circle. The resultant field is I/(2R) in magnitude. Tightly wound multiple loops of current can form an electromagnet or solenoid when the current flows.

Basic Examples


  • I = current
  • n = turns per unit length
  • R = radius of loop
  • r = shortest distance from conductor to point of measurement

1) Which is the magnitude of the magnetic field of an infinitely long straight wire at a distance r?

(a) (mu-zero/2pi) I/r

(b) (mu-zero/2) I/R

(c) (mu-zero) I n

2)Which is the magnitude of the magnetic field at the center of a circular loop of current?

(a) (mu-zero/2pi) I/r

(b) (mu-zero/2) I/R

(c) (mu-zero) I n

3)Which is the magnitude of the magnetic field on the axis of a solenoid of n turns per unit length?

(a) (mu-zero/2pi) I/r

(b) (mu-zero/2) I/R

(c) (mu-zero) I n


Two circular loops

Two identical conducting circular loops with a common center are arranged so that their planes are at right angles. What is the resultant magnetic field at the center?



Basic Examples 1(a), 2(b), 3(c)

Two Circular loops: According to the right hand (screw) rule, the magnetic fields are perpendicular to the plane of the loop. The fields add vectorially. The fields are perpendicular to each other. The resultant is the square root of 2 times the magnetic field of one of them.

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