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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

define electric image.obtain expression for electric intensity at any point on earthed conducting infinite plane due to a point charge Q placed at a distance D from it.
Mateshwar Reply
explain the lack of symmetry in the field of the parallel capacitor
Phoebe Reply
pls. explain the lack of symmetry in the field of the parallel capacitor
does your app come with video lessons?
Ahmed Reply
What is vector
Ajibola Reply
Vector is a quantity having a direction as well as magnitude
tell me about charging and discharging of capacitors
Ahemen Reply
a big and a small metal spheres are connected by a wire, which of this has the maximum electric potential on the surface.
Bundi Reply
3 capacitors 2nf,3nf,4nf are connected in parallel... what is the equivalent capacitance...and what is the potential difference across each capacitor if the EMF is 500v
Prince Reply
equivalent capacitance is 9nf nd pd across each capacitor is 500v
four effect of heat on substances
Prince Reply
why we can find a electric mirror image only in a infinite conducting....why not in finite conducting plate..?
Rima Reply
because you can't fit the boundary conditions.
what is the dimensions for VISCOUNSITY (U)
what is thermodynamics
Aniket Reply
the study of heat an other form of energy.
heat is internal kinetic energy of a body but it doesnt mean heat is energy contained in a body because heat means transfer of energy due to difference in temperature...and in thermo-dynamics we study cause, effect, application, laws, hypothesis and so on about above mentioned phenomenon in detail.
It is abranch of physical chemistry which deals with the interconversion of all form of energy
what is colamb,s law.?
Muhammad Reply
it is a low studied the force between 2 charges F=q.q`\r.r
what is the formula of del in cylindrical, polar media
Birengeso Reply
prove that the formula for the unknown resistor is Rx=R2 x R3 divided by R3,when Ig=0.
what is flux
Bundi Reply
Total number of field lines crossing the surface area
Basically flux in general is amount of anything...In Electricity and Magnetism it is the total no..of electric field lines or Magnetic field lines passing normally through the suface
what is temperature change
a bottle of soft drink was removed from refrigerator and after some time, it was observed that its temperature has increased by 15 degree Celsius, what is the temperature change in degree Fahrenheit and degree Celsius
process whereby the degree of hotness of a body (or medium) changes
where The letter "Q" is the heat transferred in an exchange in calories, "m" is the mass of the substance being heated in grams, "c" is its specific heat capacity and the static value, and "ΔT" is its change in temperature in degrees Celsius to reflect the change in temperature.
what was the temperature of the soft drink when it was removed ?
15 degree Celsius
15 degree
ok I think is just conversion
15 degree Celsius to Fahrenheit
0 degree Celsius = 32 Fahrenheit
15 degree Celsius = (15×1.8)+32 =59 Fahrenheit
I dont understand
the question said you should convert 15 degree Celsius to Fahrenheit
To convert temperatures in degrees Celsius to Fahrenheit, multiply by 1.8 (or 9/5) and add 32.
what is d final ans for Fahrenheit and Celsius
it said what is temperature change in Fahrenheit and Celsius
the 15 is already in Celsius
So the final answer for Fahrenheit is 59
what is d final ans for Fahrenheit and Celsius
what are the effects of placing a dielectric between the plates of a capacitor
Bundi Reply
increase the capacitance.
besides increasing the capacitance, is there any?
mechanical stiffness and small size
so as to increase the capacitance of a capacitor
also to avoid diffusion of charges between the two plate since they are positive and negative.
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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