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Ampere's law is a substitute expression of the Biot-Savart Law. Ampere's law (Ampere's circuit law) formula is equal to the line integral of the magnetic field around a closed loop such that it is equal to the algebraic sum of the currents passing through the loop.
- Ampere's circuital law states the relationship between a current-carrying conductor and the magnetic field formed around the conductor because of its current flow.
- Maxwell changed Ampere’s law to contain time-varying electric fields. He believed there was a displacement current existing between the plates of the capacitor. He modified the law of Ampere by having the displacement current.
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Key Terms: Ampere’s law, André-Marie Ampère, Ampere's Circuital Law, magnetic field, current, electric field
Who was André-Marie Ampère?
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André-Marie Ampère was a scientist who experimented with forces acting on current-carrying wires. This experiment was performed in the late 1820s, around the same time that Faraday was working on his law (Faraday's Law). Faraday and Ampere had no idea that Maxwell himself would combine their work four years later.
What is Ampere's Law?
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Ampere's law, also known as Ampere's circuital law, is a fundamental law of electricity and magnetism that quantifies the relationship between a magnetic field and the electric current or electric field that produces it. The law is named after the French physicist and mathematician André-Marie Ampere, who established electromagnetic theory in 1825.
- According to Ampere's law, the magnetic field created by an electric current is proportional to its magnitude. In this case, the constant of proportionality is equal to the permeability of free space.
- It also states that the magnetic field is associated with a certain current or vice versa, as long as the electric field remains constant.
- It is used to determine the magnetic induction that occurs when long current-carrying wires are used.
- It is also used to calculate the magnetic field produced by a long current-carrying cylinder.
Ampere's law is a formalized version of the Biot-Savart law, which also relates the magnetic field and the current it produces: the magnetic field line around an arbitrarily chosen path is proportional to the net electric current that the integral path encloses.
A stationary charge generates an electric field, while a moving charge generates a magnetic field. Ampere's law is a mathematical statement that describes the link between the current and magnetic field created in electromagnetism. An electric current produces a magnetic field that is proportional to the magnitude of the current, which is proportional to the permeability of free space (μ0), a universal constant in physics. Its value is 4π x 10−7H/m.
In its different form, Ampere's law states that the sum of the elements of the length of the magnetic field in the direction of the element of length is equal to the coefficient of permeability of the electric current enclosed in the loop for any closed channel (Ampere's loop/ Amperian loop).
Both integral and differential formulations of the law of Ampere are reasonable. In most cases, the integral form is used to calculate the magnetic field. This magnetic field will give the same result if it is derived from the Biot-Savart law.
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Ampere's Law Equations
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Ampere's law states that the magnetic field produced by an electric current is proportional to its magnitude, which is proportional to the permeability of free space.
The magnetic field due to a current-carrying wire can be calculated using Biot-Savart's law, from which Ampere's law can be derived. According to Biot-Savart's law, the magnetic field produced by a long straight wire is
B = \(\frac{\mu_0I}{2 \pi r}\)
As \(\vec{B}\) and \(\vec{dl}\) are in a similar direction,
\(\vec{B}\) × \(\vec{dl}\) = B dl cos0 = B dl
∴ ∮ \(\vec{B}\) × \(\vec{dl}\) = \(\frac{\mu_0I}{2 \pi r}\) ∮ dl
∮ \(\vec{B}\) × \(\vec{dl}\) = \(\frac{\mu_0I}{2 \pi r}\) ∮ (2πr)
∮ \(\vec{B}\) × \(\vec{dl}\) = μ0I
where I = electric current,
\(\vec{B}\) = magnetic field, and
μ0 = permeability of free space.
Ampere's Circuital Law Checkpoints and Conditions
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According to Ampere's circuital equation, the line integral of the magnetic field around a closed loop is equal to the algebraic sum of the current passing through the loop.
∮H.dL = Ienc
If the conductor carries a current I, the current flow creates a magnetic field around the wire. The left side of the equation states that if an imaginary path surrounds the wire and a magnetic field is applied at each location, the current enclosed by this path, denoted by Ienc, is numerically equivalent to the current ringed by this path.
The magnetic field is calculated using the integral form of Ampere's law, which can be integrated over space. Consequently, it is used to determine the fields produced by devices such as long straight conducting wires, cylindrical conductors, coaxial cables, solenoids, and toroids. The right-hand rule of thumb is commonly used to determine the direction of a magnetic field.
Magnetic field because of a long straight wire (from the law of Ampere):
∮ \(\vec{B}\) x \(\vec{dl}\) = μ0I
As \(\vec{B}\) and \(\vec{dl}\) are in the similar direction,
\(\vec{B}\) x \(\vec{dl}\) = B dl cos0 = B dl
∴ ∮ \(\vec{B}\) x \(\vec{dl}\) = B ∮ dl (2πr)
Therefore, \(\vec{B}\) (2πr) = μ0I
B = \(\frac{\mu_0I}{2 \pi r}\)
A magnetic field caused by a current-carrying wire can be caused by straight wires, circular coils, and solenoids.
When current flows in a straight wire, the magnetic field produced consists of circular lines of force, centered on the wire.
The plane of the circular lines is parallel to the length of the wire. Arrows indicate their direction.
To determine the direction of the current, you can use the right-hand rule.
The direction the fingers turn indicates the direction in which the magnetic field lines travel around the wire.
Example: If the current in the wire is going north, imagine that the thumb is pointing north. When the fingers turn, they may go under the wire to the left and up to the right of the wire, or under the wire to the west and up to the east.
Applications of Ampere’s Law
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The following are the applications of Ampere’s Law:
- With the help of Ampere’s Law, it is possible to identify the magnetic induction because of the long current-carrying wire.
- Ampere’s Law also helps in identifying the magnetic field inside the toroid and inside the conductor.
- The magnetic field produced by a cylinder carrying a long current can also be determined with the help of Ampere’s law.
- Ampere’s Law also helps in identifying the forces between the currents.
Also Read:
| Related Articles | ||
|---|---|---|
| Magnetic Spectrum | Faraday's Law of Electromagnetic Induction | Gauss’s Law |
Things to Remember
- André-Marie Ampère performed an experiment with forces acting on current-carrying wires in the late 1820s.
- Ampere's law (Ampere's circuital law) is a basic law of electricity and magnetism quantifying the relationship between a magnetic field and the electric current or electric field that generates it.
- Ampere's Law Equations:
B = \(\frac{\mu_0I}{2 \pi r}\)
∮ \(\vec{B}\) x \(\vec{dl}\) = μ0I
- With the help of Ampere’s Law, it is possible to Identify:
- The magnetic induction because of the long current-carrying wire.
- The magnetic field inside the toroid as well as inside the conductor.
- The magnetic field produced by a cylinder carrying a long current.
- The forces between the currents.
Sample Questions
Ques. Calculate the magnetic field of a long straight wire having a circular loop with a radius of 0.05m. The reading of the current flowing via this closed loop is 2amp. (3 marks)
Ans. Given
R = 0.05m
I = 2amp
μ0 = 4π × 10-7 N/A2
Ampere’s law formula is
∮ \(\vec{B}\) x \(\vec{dl}\) = μ0I
In the case of long straight wire
∮\(\vec{dl}\) = 2πR
= 3.14 × 2 × 0.05
= 0.314
\(\vec{B}\) ∮ \(\vec{dl}\) = μ0I
\(\vec{B}\) = \(\frac{\mu_0I}{2 \pi r}\)
\(\vec{B}\) = 4π ×10-7 / 20.314 = 8 × 10-6 T
Ques. If a current of 2 A is flowing through a closed wire having a radius of 0.2 m, calculate its magnetic field. (3 marks)
Ans. We have,
r = 0.2
I = 2
μ0 = 4π ×10-7
For this, the length of the loop will be,
∮\(\vec{dl}\) = 2πR
= 2 (22/7) (0.2)
= 1.25 m
Using the formula we have,
\(\vec{B}\) = \(\frac{\mu_0I}{2 \pi r}\)
= (4π × 10-7) (2) / (1.25)
= 2.011 × 10-6 T
Ques. Calculate the magnetic field if a current of 3 A is flowing through a closed wire having a radius of 0.5 m. (3 marks)
Ans. We have,
r = 0.5
I = 3
μ0 = 4π × 10-7
For this, the length of the loop will be,
∮\(\vec{dl}\) = 2πR
= 2 (22/7) (0.5)
= 6.28 m
Using the formula we have,
\(\vec{B}\) = \(\frac{\mu_0I}{2 \pi r}\)
= (4π × 10-7) (3) / (6.28) = 6 × 10-7 T
Ques. Calculate the magnetic field of a closed wire having a radius of 0.8 m if a current of 5 A is flowing through the same. (3 marks)
Ans. We have,
r = 0.8
I = 5
μ0 = 4π × 10-7
For this, the length of the loop will be,
∮\(\vec{dl}\) = 2πR
= 2 (22/7) (0.8)
= 5.02 m
Using the formula we have,
\(\vec{B}\) = \(\frac{\mu_0I}{2 \pi r}\)
= (4π × 10-7) (5) / (5.02) = 1.25 × 10-6 T
Ques. A current of 10 A is flowing through a closed wire having a radius of 0.4 m. Calculate its magnetic field. (3 marks)
Ans. We have,
r = 0.4
I = 10
μ0 = 4π × 10-7
In our case, the length of the loop will be,
∮\(\vec{dl}\) = 2πR
= 2 (22/7) (0.4)
= 2.51 m
Using the formula we have,
\(\vec{B}\) = \(\frac{\mu_0I}{2 \pi r}\)
= (4π ×10-7) (10) / (2.51) = 5 × 10-6 T
Ques. The field of a closed wire is 3.4 × 10-6 T. Calculate the current flowing through it if its radius is 0.7 m. (3 marks)
Ans. We have,
r = 0.7
B = 3.4 × 10-6
μ0 = 4π × 10-7
In our case, the length of the loop will be,
∮\(\vec{dl}\) = 2πR
= 2 (22/7) (0.7)
= 4.4 m
Using the formula we have,
\(\vec{B}\) = \(\frac{\mu_0I}{2 \pi r}\)
3.4 × 10-6 = (4π × 10-7) (I) / (2.51)
I = 85.3 / 12.57 = 6.78 A
Ques. Calculate the current flowing via a closed wire of a radius of 0.32 m if the field of the same is 2.76 × 10-7 T. (3 marks)
Ans. We have,
r = 0.32
B = 2.76 × 10-7
μ0 = 4π × 10-7
In our case, the length of the loop will be,
∮\(\vec{dl}\) = 2πR
= 2 (22/7) (0.32)
= 2.011 m
Using the formula we have,
\(\vec{B}\) = \(\frac{\mu_0I}{2 \pi r}\)
2.76 × 10-7 = (4π ×10-7) (I) / (2.011) = 5.55 / 12.57 = 0.44 A
Ques. The field of a closed wire is 8.21 × 10-5 T and the flow of current is 7 A. Calculate its radius. (3 marks)
Ans. We have,
B = 8.21 × 10-5
μ0 = 4π × 10-7
I = 7
Using the formula we have,
\(\vec{B}\) = \(\frac{\mu_0I}{2 \pi r}\)
8.21 × 10-5 = (4π × 10-7) (7) / r
r = (87.99 × 10-2) / 8.21 = 0.017 m
Ques. Find the current in a long straight wire that would produce a magnetic field twice the strength of the Earth’s at a distance of 5.0cm from the wire. (Magnetic field of Earth = 5.0 × 10−5 T, μ0 = 4π × 10−7 Tm/A) (3 marks)
Ans. According to the question,
Bwire = 2 ⨯ Bearth
∴ Bwire = 10 ⨯ 10-5 T = 1⨯10-4 T
r = 5.0 cm
We know,
I = 2πrB / μ0
= 2π (5.0 × 10-2) (1 × 10-1 T) / 4π × 10-7 Tm/A
∴I = 25A
Ques: A finite-length conductor has a current of 5A and a radius of 15cm in the air (in 10-6 order). What is its magnetic flux density? (3 marks)
Ans. Given,
I = 5
AR = 15 cm
As,
B = mu0I / 2πr
= 4π × 10-7 × 3 / 2π × 0.12 = 5 × 10-6 T
Ques. Mark the incorrect option. (1 mark)
(A) Ampere's law states that the current B through any closed surface is μo times the current through the area enclosed by the closed surface.
(B) Gauss’s law of magnetic field performs the same meaning as the law of Gauss for the electric field.
(C) Gauss's magnetic field law states that the flux B in any closed surface is zero, regardless of whether there is any current inside the surface.
(D) All of the above.
Ans. Option A. Ampere law expresses that for any close-looped route, the total of the length elements times the magnetic field in the direction of the length element is similar to the permeability times the electric current surrounded in the loop.
Ques. A student gets confused about whether two parallel wires carrying current in a similar direction attract or repel. What rules would he require to get the correct conclusion? (1 mark)
(A) Fleming Heft Hand Rule
(B) Right-Hand Thumb Rule
(C) Both A and B
(D) None
Ans. C. Suppose two parallel wires have current in a similar direction. When the right-hand rule of thumb and Fleming's left-hand rule are applied, it is seen that the force in the direction of the first wire means that the second string is attracted to the second string. Likewise, the 1st wire also attracts the 2nd wire. Therefore, they attract.
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