<< Chapter < Page Chapter >> Page >
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

I'm interested in biological psychology and cognitive psychology
Tanya Reply
what does preconceived mean
sammie Reply
physiological Psychology
Nwosu Reply
How can I develope my cognitive domain
Amanyire Reply
why is communication effective
Dakolo Reply
Communication is effective because it allows individuals to share ideas, thoughts, and information with others.
effective communication can lead to improved outcomes in various settings, including personal relationships, business environments, and educational settings. By communicating effectively, individuals can negotiate effectively, solve problems collaboratively, and work towards common goals.
it starts up serve and return practice/assessments.it helps find voice talking therapy also assessments through relaxed conversation.
miss
Every time someone flushes a toilet in the apartment building, the person begins to jumb back automatically after hearing the flush, before the water temperature changes. Identify the types of learning, if it is classical conditioning identify the NS, UCS, CS and CR. If it is operant conditioning, identify the type of consequence positive reinforcement, negative reinforcement or punishment
Wekolamo Reply
please i need answer
Wekolamo
because it helps many people around the world to understand how to interact with other people and understand them well, for example at work (job).
Manix Reply
Agreed 👍 There are many parts of our brains and behaviors, we really need to get to know. Blessings for everyone and happy Sunday!
ARC
A child is a member of community not society elucidate ?
JESSY Reply
Isn't practices worldwide, be it psychology, be it science. isn't much just a false belief of control over something the mind cannot truly comprehend?
Simon Reply
compare and contrast skinner's perspective on personality development on freud
namakula Reply
Skinner skipped the whole unconscious phenomenon and rather emphasized on classical conditioning
war
explain how nature and nurture affect the development and later the productivity of an individual.
Amesalu Reply
nature is an hereditary factor while nurture is an environmental factor which constitute an individual personality. so if an individual's parent has a deviant behavior and was also brought up in an deviant environment, observation of the behavior and the inborn trait we make the individual deviant.
Samuel
I am taking this course because I am hoping that I could somehow learn more about my chosen field of interest and due to the fact that being a PsyD really ignites my passion as an individual the more I hope to learn about developing and literally explore the complexity of my critical thinking skills
Zyryn Reply
good👍
Jonathan
and having a good philosophy of the world is like a sandwich and a peanut butter 👍
Jonathan
generally amnesi how long yrs memory loss
Kelu Reply
interpersonal relationships
Abdulfatai Reply
What would be the best educational aid(s) for gifted kids/savants?
Heidi Reply
treat them normal, if they want help then give them. that will make everyone happy
Saurabh
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
Practice Key Terms 2

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, University physics volume 2. OpenStax CNX. Oct 06, 2016 Download for free at http://cnx.org/content/col12074/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'University physics volume 2' conversation and receive update notifications?

Ask