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By the end of this section, you will be able to:
  • Explain how to derive a magnetic field from an arbitrary current in a line segment
  • Calculate magnetic field from the Biot-Savart law in specific geometries, such as a current in a line and a current in a circular arc

We have seen that mass produces a gravitational field and also interacts with that field. Charge produces an electric field and also interacts with that field. Since moving charge (that is, current) interacts with a magnetic field, we might expect that it also creates that field—and it does.

The equation used to calculate the magnetic field produced by a current is known as the Biot-Savart law. It is an empirical law named in honor of two scientists who investigated the interaction between a straight, current-carrying wire and a permanent magnet. This law enables us to calculate the magnitude and direction of the magnetic field produced by a current in a wire. The Biot-Savart law    states that at any point P ( [link] ), the magnetic field d B due to an element d l of a current-carrying wire is given by

d B = μ 0 4 π I d l × r ^ r 2 .
This figure demonstrates Biot-Savart Law. A current dI flows through a magnetic wire. A point P is located at the distance r from the wire. A vector r to the point P forms an angle theta with the wire. Magnetic field dB exists in the point P.
A current element I d l produces a magnetic field at point P given by the Biot-Savart law.

The constant μ 0 is known as the permeability of free space    and is exactly

μ 0 = 4 π × 10 −7 T m/A

in the SI system. The infinitesimal wire segment d l is in the same direction as the current I (assumed positive), r is the distance from d l to P and r ^ is a unit vector that points from d l to P , as shown in the figure.

The direction of d B is determined by applying the right-hand rule to the vector product d l × r ^ . The magnitude of d B is

d B = μ 0 4 π I d l sin θ r 2

where θ is the angle between d l and r ^ . Notice that if θ = 0 , then d B = 0 . The field produced by a current element I d l has no component parallel to d l .

The magnetic field due to a finite length of current-carrying wire is found by integrating [link] along the wire, giving us the usual form of the Biot-Savart law.

Biot-savart law

The magnetic field B due to an element d l of a current-carrying wire is given by

B = μ 0 4 π wire I d l × r ^ r 2 .

Since this is a vector integral, contributions from different current elements may not point in the same direction. Consequently, the integral is often difficult to evaluate, even for fairly simple geometries. The following strategy may be helpful.

Problem-solving strategy: solving biot-savart problems

To solve Biot-Savart law problems, the following steps are helpful:

  1. Identify that the Biot-Savart law is the chosen method to solve the given problem. If there is symmetry in the problem comparing B and d l , Ampère’s law may be the preferred method to solve the question.
  2. Draw the current element length d l and the unit vector r ^ , noting that d l points in the direction of the current and r ^ points from the current element toward the point where the field is desired.
  3. Calculate the cross product d l × r ^ . The resultant vector gives the direction of the magnetic field according to the Biot-Savart law.
  4. Use [link] and substitute all given quantities into the expression to solve for the magnetic field. Note all variables that remain constant over the entire length of the wire may be factored out of the integration.
  5. Use the right-hand rule to verify the direction of the magnetic field produced from the current or to write down the direction of the magnetic field if only the magnitude was solved for in the previous part.

Questions & Answers

Why does the lines of force not touch each other 🇲🇲
Gbemisola Reply
what is unit
Rayyanu Reply
Please canu get more questions on electric field and electric flux please
Gbemisola
is electric field directly proportional to the squared of a distance
Benjamin Reply
No electric field is inversely proportional to the squared distance between the charges
Gbemisola
lets treat linear expansivity please
Ujah Reply
The bullet 2.00cm long is fired at 420/s and passes straight through a 10.0 cm thick board existing at 280 m/s.What is the average acceleration of the bullet through the board?
FAUSTINA
an unstretched spring is 12cm long .A load of 5N stretched it to 15cm .how long will it be under a load of 15N?
Morapeli
hi
Africa
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Benjamin
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like 100 level
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Africa
Compare the electric flux through the surface of a cube of side length a that has a charge q at its center to the flux through a spherical surface of radius a with a charge q at its center.
Shari Reply
please I want to know how to solve increase in length
Ujah
Why a charged capacitor has potential difference but not emf
Gideon Reply
what is the dimension symbol of temperature?
Keren Reply
what is the dimension symbol of temperature?
Keren
what's the meaning of enthalpy in terms of latent heat, internal energy, phase change
Anthony Reply
Enthalpy is the degree ofdisorderlinessof a substance
Gbemisola
how to convert Kelvin to centigrade
Sangeetha Reply
what is the s, p, d, f in this table
Sangeetha
s, p, d, f in this table
Sangeetha
what kind of table this
Vengata
Periodic table
Gbemisola
what are waves
KENNETH Reply
In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities
Abdikadir
Discuss how would orient a planar surface of area A in a uniform electric field of magnitude E0 to obtain (a) the maximum flux and (b) the minimum flux through the area.
KARAN Reply
I'm just doing the first 3 with this message. but thankyou for the time your obviously intending to support us with. viva la accumulation
Marcel
Find the net capacitance of the combination of series and parallel capacitors shown belo
jean Reply
what is ohm?
Sharafat Reply
calculate ideal gas pressure of 0.300mol,v=2L T=40°c
Viola Reply
what is principle of superposition
Jyoti Reply
Practice Key Terms 2

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Source:  OpenStax, University physics volume 2. OpenStax CNX. Oct 06, 2016 Download for free at http://cnx.org/content/col12074/1.3
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