Ampere's law is one of the standard laws of physics. According to it, "the magnetic field formed by an electric current is proportional to the magnitude of the current passing through the conductor and the constant of proportionality is equal to the permeability of space."
Ampere law is the fundamental law that provides a Relation Between the Current and the Magnetic field around it. Let's learn more about Ampere's law with its Application in detail in this article.
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André-Marie Ampère
The famous French physicist André-Marie Ampère was the first to provide the relationship between the current and the magnetic field produced by it. He performed various experiments on current-carrying wires and studied the force that is applied to them. He was a contemporary of Michael Faraday (who gives Faraday's Law of Electromagnetism). His law proves t be fundamental in learning modern-day electromagnetics.
Ampere’s Law Definition
Ampere’s law is one of the fundamental laws of electrostatics which states that the electric current in the conductor produced the magnetic field.
Ampere’s Law states that,
“The magnetic field produced by an electric current is directly proportional to the intensity of the electric current and the constant of probability here is the permeability of free space."
The Maxwell equation explaining Ampere's law is given below,
Ampere’s Circuital Law Definition
Ampere's Circuital law state that the line integral of the magnetic field in a closed loop gives the current passing through the loop.
The following image shows the Ampere’s Circuital formula,
Ampere's Law Formula
The formula for Ampere's law is equal to the line integral of the magnetic field around a closed loop such that it equals the number of times the algebraic sum of currents is passing through the loop. For a conductor carrying current 'I' such that the flow of current creates a magnetic field around the wire, this formula can be used to calculate the field.
\oint\vec{B}.\vec{dl}=μ_oI
Notations Used In Ampere’s Law Formula
- μo is the permeability constant with a value of 4π × 10-7 N/A2
- B is the magnetic field
- I is the flow of current passing through the closed loop
- L is the length of the loop
For a closed wire, the value of
B = μoI/2πr
Determining Magnetic Field by Ampere’s Law
The magnetic field at r distance from the wire is calculated by using Ampere’s Law. For a wire conducting I current the magnetic field at distance r is calculated using Ampere's law and its direction is given using Right Hand Thumb Rule.
For calculating the magnetic field around the wire we draw an imaginary route at a distance r from the wire. From the second Maxwell equation, the magnetic field integrated along this path gives the current enclosed in the wire, i.e. I.
The magnetic field does not vary with the change in the distance r. The length of the path covered is 2πr. Now using Ampere's Law the magnetic field is given using the formula discussed in the image below,
Applications of Ampere’s Law
Various applications of Ampere’s Law are,
- It is used to calculate the magnetic field produced by the current-carrying wire.
- Ampere’s Law gives a magnetic field inside a toroid
- Ampere’s Law gives Magnetic Field inside the conductor
- It also provides ways to calculate the force between two conductors.
Solved Examples on Ampere’s Law
Example 1: Find the magnetic field of a closed wire with a radius of 0.2 m if a current of 2 A is flowing through it.
Solution:
We have,
r = 0.2 m
I = 2 A
μo = 4π × 10-7In our case, the length of loop is,
\oint\vec{dl} = 2πr= 2 (22/7) (0.2)
= 1.25 m
Using the formula we have,
B = μoI/2πr
= (4π × 10-7) (2)/(1.25)
= 2.011 × 10-6 T
Example 2: Find the magnetic field of a closed wire with a radius of 0.5 m if a current of 3 A is flowing through it.
Solution:
We have,
r = 0.5 m
I = 3 A
μo = 4π × 10-7In our case, the length of loop is,
\oint\vec{dl} = 2πr= 2 (22/7) (0.5)
= 6.28 m
Using the formula we have,
B = μoI/2πr
= (4π × 10-7) (3)/(6.28)
= 6 × 10-7 T
Example 3: Find the magnetic field of a closed wire with a radius of 0.8 m if a current of 5 A is flowing through it.
Solution:
We have,
r = 0.8 m
I = 5 A
μo = 4π × 10-7In our case, the length of loop is,
\oint\vec{dl} = 2πr= 2 (22/7) (0.8)
= 5.02 m
Using the formula we have,
B = μoI/2πr
= (4π × 10-7) (5)/(5.02)
= 1.25 × 10-6 T
Example 4: Find the magnetic field of a closed wire with a radius of 0.4 m if a current of 10 A is flowing through it.
Solution:
We have,
r = 0.4 m
I = 10 A
μo = 4π × 10-7In our case, the length of loop is,
\oint\vec{dl} = 2πr= 2 (22/7) (0.4)
= 2.51 m
Using the formula we have,
B = μoI/2πr
= (4π × 10-7) (10)/(2.51)
= 5 × 10-6 T
Example 5: Find the current flowing through a closed wire of a radius of 0.7 m if its field is 3.4 × 10-6 T.
Solution:
We have,
r = 0.7 m
B = 3.4 × 10-6 T
μo = 4π × 10-7In our case, the length of loop is,
\oint\vec{dl} = 2πr= 2 (22/7) (0.7)
= 4.4 m
Using the formula we have,
B = μoI/2πr
3.4 × 10-6 = (4π × 10-7) (I)/(2.51)
I = 85.3/12.57
I = 6.78 A
Example 6: Find the current flowing through a closed wire of a radius of 0.32 m if its field is 2.76 × 10-7 T.
Solution:
We have,
r = 0.32 m
B = 2.76 × 10-7 T
μo = 4π × 10-7In our case, the length of loop is,
\oint\vec{dl} = 2πr= 2 (22/7) (0.32)
= 2.011 m
Using the formula we have,
B = μoI/2πr
2.76 × 10-7 = (4π × 10-7) (I)/(2.011)
I = 5.55/12.57
I = 0.44 A
Example 7: Find the radius of a closed wire if its field is 8.21 × 10-5 T and the current flow is 7 A.
Solution:
We have,
B = 8.21 × 10-5 A
μo = 4π × 10-7
I = 7 AUsing the formula we have,
B = μoI/2πr
8.21 × 10-5 = (4π × 10-7) (7)/r
r = (87.99 × 10-2)/8.21
r = 0.017 m
