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Ampere's Circuital Law And Its Applications

Ampere's Circuital Law And Its Applications

Edited By Vishal kumar | Updated on Jul 02, 2025 05:49 PM IST

Imagine winding a wire around a nail and making an electromagnet. How would you calculate the magnetic field around the wire? Ampere's circuital law creates a relation between the flowing electric current and the consequent magnetic field produced. It forms one of the base principles of electromagnetism, critical for designing electric motors and transformers, among many others. Let's see how Ampere's Circuital Law can read the output of a wound circuit amongst other things in electronics and magnetism.

This Story also Contains
  1. Ampers Circitue Law
  2. Magnetic Field Due to Infinitely Long Cylindrical Wire Due to Current 'i' Distributed Uniformly Across Its Cross-Section
  3. Solved Example Based On Ampere's Circuital Law And Its Applications
  4. Summary

Further, we discuss in this article, Ampere's Circuital Law And Its Applications is important for board exams and comparative exams. This concept is in the class 12th Magnetic Effects of Current and Magnetism chapter. Questions asked from these competitive exams like the JEE Main, NEET, and other entrance exams such as SRMJEE, BITSAT, WBJEE, BCECE and more. Over the last ten years of the JEE Main exam (from 2013 to 2023), a total of seventeen questions have been asked on this concept. And for NEET five questions were asked from this concept.

Ampers Circitue Law

Amperes law is also a method to calculate the magnetic field due to a given current distribution like Biot-Savart's law.

Statement: The line integral of the magnetic field $\vec{B}$ around any closed curve is equal to $\mu_0$ times the total current i passing through the area enclosed by the curve.

Mathematical statement: $\oint \vec{B} d \vec{l}=\mu_0 \sum i=\mu_0\left(i_1+i_3-i_2\right)$

Also using $\vec{B}=\mu_0 \vec{H} \quad \ldots($ where $\vec{H}=$ magnetising field $)$
$
\oint \mu_0 \vec{H} \cdot \overrightarrow{d l}=\mu_0 \Sigma i \Rightarrow \oint \vec{H} \cdot \overrightarrow{d l}=\Sigma i
$

Fingers are curled in the loop direction, the current in the direction of the thumb is taken as positive whereas in the direction opposite to that of the thumb is taken as negative.

Now, we can see that the total current crossing the above area is $\left(i_1+i_3-i_2\right)$, so any current outside the given area will not be considered. So we have to assume

(Outward $\odot \rightarrow+v e$, Inward $\otimes \rightarrow-v e$ )

General Guidelines for the Selection of Ampere's Path for its Application in Different Situations

(i) Path should be chosen in such a way that at every point of the path magnetic induction should be either tangential to the path elements or normal to it so that the 'dot' product can be easily handled.

(ii) Path should be chosen in such a way that at every point of the path magnetic induction should either be uniform or zero so that calculations become easy.

Application of Ampere's law

Magnetic field due to infinite straight wire -

As by Biot-savant law, we have calculated the magnetic field. Now with the help of Ampere's circuital law, let us take an infinite straight wire, now if we want to calculate the magnetic field of this wire at a distance of 'd' from the wire. Then,

$\oint \vec{B} \cdot \overrightarrow{d l}=\mu_o \cdot i_{\text {inside }}$

See the figure given below, it represents the magnetic field produced by the current-carrying wire and the length vector. We can see that the ' B ' and 'dl' are making an angle 0 degree, so the $\operatorname{Cos} \theta=1$

Also, B is constant as the current is constant which is flowing through the wire. So we can write that -

$B \oint d l=\mu_o \cdot i$

Now the length of the loop is the circumference of the loop. Here the radius is 'd' and full integration of 'dl' is equal to $2 \pi d$. So we can write that -

$\begin{aligned} & B .2 \pi d=\mu_o i \\ & B=\frac{\mu_o i}{2 \pi d}\end{aligned}$

One can notice that the result obtained is the same as that we have obtained by Biot-savant law.

Magnetic Field Due to Infinitely Long Cylindrical Wire Due to Current 'i' Flowing Through Its Surface Only

Let us consider an infinitely long cylindrical wire of radius R and the current is distributed on the surface of the wire, then this wire will behave as a hollow cylindrical wire.

Now let us take different situations -

a) For a point inside the wire - (r<R)

From the top view, the Ampere's loop will look like this -

Since there is no current inside the Ampere's loop, so there will be no magnetic field in this loop because -

$\oint \vec{B} \cdot \overrightarrow{d l}=\mu_0 i$

b) For a point outside the wire (r>R) -

Then from the top, it can be seen as -

It is just like the concept of a current carrying wire which we have studied in the last concept with the help of Ampere's circuital law as well as by Biot-savart law. So again by applying same Ampere's circuital law we can deduce that -

$
B=\frac{\mu_o i}{2 \pi r}
$
c) On the surface $(\mathrm{r}=\mathrm{R})$
$
B_s=\frac{\mu_o i}{2 \pi r}
$

From the above equations, we can plot a graph between B and different positions 'r'.

Magnetic Field Due to Infinitely Long Cylindrical Wire Due to Current 'i' Distributed Uniformly Across Its Cross-Section

Magnetic field due to a cylindrical wire is obtained by the application of Ampere's law. Here also we consider few cases one by one -

a) Outside the cylinder -

It is just like the concept of a current carrying wire which we have studied in the last concept with the help of Ampere's circuital law as well as by Biot-savart law. So again by applying same Ampere's circuital law we can deduce that -

$B=\frac{\mu_o i}{2 \pi r}$

b) Inside the solid cylinder : Current enclosed by loop (i') is lesser then the total current (i) -

Since the current density will remain the same.

So,

$
J=J^{\prime} \Rightarrow \quad i=i \times \frac{A^{\prime}}{A}=i\left(\frac{r^2}{R^2}\right)
$

Hence at inside point $\oint \overrightarrow{B_{\text {in }}} \cdot d \vec{l}=\mu_0 i \Rightarrow B=\frac{\mu_0}{2 \pi} \cdot \frac{i r}{R^2}$

c) At surface (r=R) -

$B_s=\frac{\mu_o i}{2 \pi r}$

The variation of B with r can be drawn as -

Solved Example Based On Ampere's Circuital Law And Its Applications

Example 1: Two wires with current 2A and 1A are enclosed in a circular loop. Another wire with current 3A is situated outside the circular loop. The $\int B \cdot \overrightarrow{d l}$ around the loop is:

1) $\mu_0$
2) $3 \mu_0$
3) $6 \mu_0$
4) $2 \mu_0$

Solution:

As we learned

Ampere's Law -

It gives another method to calculate the magnetic field due to the given current distribution.

Another wire with current 3A is situated outside the loop as shown-

According to Ampere's circuital law

$\oint \vec{B} \cdot \overrightarrow{d l}=\mu_0 I_{\text {enclosed }}=\mu_0(2 A-1 A)=\mu_0$

Example 2: A current I flows along the length of an infinitely long straight and thin-walled pipe then

1) $\vec{B}$ at all points inside the pipe is the same but non-zero
2) $\vec{B}$ at any point inside the pipe is zero
3) $\vec{B}$ is zero only at the axis of the pipe
4) $\vec{B}$ is different at different points

Solution:

As we learned

Ampere's Law

It gives another method to calculate the magnetic field due to the given current distribution.

Amperian loop of radius r inside cylinder

Applying Ampere's law
For closed path.
$\mathrm{I}=0$ inside pipe
So $\vec{B}=0$ for every point

Example 3: A large metal sheet carries an electric current along its surface current per unit length is $\lambda$. Magnetic field induction near the metal surface is

1) $\lambda \mu o$
2) $\frac{\lambda \mu o}{2}$
3) $\frac{\lambda \mu o}{2 \pi}$
4) $\frac{\mu o}{2 \pi \lambda}$

Solution:

As we learned

The magnetic field in the Infinite sheet carrying current -

$
B=\frac{\mu_o j}{2}
$
$J \rightarrow$ current density

- wherein

Use ampere circuital law

$\begin{aligned} & \oint \vec{B} \cdot \overrightarrow{d l}=\mu o I \\ & 2 B l=\mu o(\lambda l) \\ & B=\frac{\mu o \lambda}{2}\end{aligned}$

Example 4: The figure shows an amperian path $A B C D A$ Part $A B C$ is in the vertical plane PSTU while part $C D A$ is in the horizontal plane $P Q R S$. The direction of circulation along the path is shown by an arrow near points $B$ and $D$.

$\oint \vec{B} \cdot \overrightarrow{d l}$ for this path according to Ampere’s law will be :

1) $\left(i_1-i_2+i_3\right) \mu_0$
2) $\left(-i_1+i_2\right) \mu_0$
3) $i_3 \mu_0$
4) $\left(i_1+i_2\right) \mu_0$

Solution:

$\oint_{A B C D A} \vec{B} \overrightarrow{d l}=\mu_0\left(i_1+i_3+i_2-i_3\right)=\mu_0\left(i_1+i_2\right)$

[Since for the given direction of circulation $i_3$ entering at PSTU is positive while $i_3$ at PQRS is negative]

Example 5: A long-walled pipe of radius R carries a current i along its length. The current density is uniform over the circumference of the pipe. The magnetic field at the centre of the pipe due to the quarter portion of the pipe shown, is

1) $\frac{\mu_0 i \sqrt{2}}{4 \pi^2 R}$
2) $\frac{\mu_{0 \mathrm{i}}}{\pi^2 R}$
3) $\frac{2 \mu_0 i \sqrt{2}}{\pi^2 R}$
4) None

Solution:

$\begin{aligned} & \lambda=\frac{i}{2 \pi r} \\ & d i=\lambda \cdot R d \theta \\ & d B=\frac{\mu_0 \lambda R d \theta}{2 \pi R} \\ & B_{\perp}=\int_0^{\pi / 2} d B \sin \theta=\frac{\mu_0 \lambda}{2 \pi}=\frac{\mu_0 i}{4 \pi^2 R} \\ & B_{11}=\int_0^{\pi / 2} d B \sin \theta=\frac{\mu_0 i}{4 \pi^2 R} \\ & B_{\text {net }}=\sqrt{2} \frac{\mu_0 i}{4 \pi^2 R} \quad\left(\because B_{\text {net }}=\sqrt{\left(B_{\perp}\right)^2+\left(B_{11}\right)^2}\right)\end{aligned}$

Summary

Ampere's Circuital Law says that a magnetic field in space around the electric current is proportional to the current itself. In general, it relates the integrated magnetic field around a closed loop to the electric current passing through the loop. It is expressed as the integral of the magnetic field around a closed path equating to the permeability of free space times the current enclosed by the path. In this law wide applications in the design of electromagnets, analysis of magnetic field in electrical circuits, and study of inductor and transformer behaviours.

Frequently Asked Questions (FAQs)

1. What is Ampere's Circuital Law?
Ampere's Circuital Law states that the line integral of magnetic field around any closed loop is equal to the product of the permeability of free space and the total current enclosed by the loop. It relates the magnetic field around a closed path to the electric current passing through the path.
2. How does Ampere's law differ from Biot-Savart law?
While both laws describe magnetic fields, Ampere's law is more general and applies to any closed loop, whereas Biot-Savart law is used for calculating the magnetic field due to a current element. Ampere's law is often easier to use for symmetric current distributions, while Biot-Savart law is more versatile for asymmetric situations.
3. Why is the direction of the amperian loop important in Ampere's law?
The direction of the amperian loop determines the sign of the current enclosed. By convention, we use the right-hand rule: if the fingers of the right hand curl in the direction of integration around the loop, the thumb points in the positive direction for current. This ensures consistency in calculations.
4. Can Ampere's law be applied to non-steady currents?
Ampere's law in its original form applies only to steady currents. For time-varying currents, Maxwell modified the law by adding a displacement current term. This modified version, known as the Ampere-Maxwell law, is valid for both steady and non-steady currents.
5. What is the significance of the permeability of free space in Ampere's law?
The permeability of free space (μ₀) in Ampere's law represents how easily a magnetic field can be established in vacuum. It acts as a proportionality constant relating the current to the magnetic field strength, and its value is 4π × 10⁻⁷ T·m/A in SI units.
6. How does Ampere's law apply to a long, straight wire?
For a long, straight wire, Ampere's law simplifies to B = μ₀I / (2πr), where B is the magnetic field strength at a distance r from the wire, and I is the current in the wire. This results in circular magnetic field lines around the wire, with field strength decreasing with distance.
7. Can Ampere's law be used to find the magnetic field inside a wire carrying current?
Yes, Ampere's law can be used to find the magnetic field inside a current-carrying wire. Inside the wire, only the current enclosed by the amperian loop at that radius contributes to the field. This results in a magnetic field that increases linearly from zero at the center to a maximum at the surface.
8. What is the importance of symmetry in applying Ampere's law?
Symmetry is crucial in applying Ampere's law because it allows us to simplify the problem. In symmetric situations (like long straight wires or solenoids), we can choose amperian loops where the magnetic field is constant or has a simple relationship to position, making the integration much easier.
9. How does Ampere's law relate to the Biot-Savart law in terms of magnetic field calculations?
Both laws can be used to calculate magnetic fields, but they're suited for different situations. Ampere's law is often easier for highly symmetric current distributions, while the Biot-Savart law is more versatile for arbitrary current distributions. Mathematically, Ampere's law can be derived from the Biot-Savart law.
10. How does Ampere's law apply to coaxial cables?
For a coaxial cable, Ampere's law can be used to find the magnetic field between the inner and outer conductors. The field is similar to that of a long straight wire, but only the current in the inner conductor contributes, as the net current enclosed by any loop between the conductors is constant.
11. What is the significance of the right-hand rule in Ampere's law?
The right-hand rule in Ampere's law helps determine the direction of the magnetic field. If you wrap your right hand around the current-carrying conductor with your thumb pointing in the direction of current flow, your fingers will curl in the direction of the magnetic field lines.
12. What is the limitation of Ampere's law in calculating magnetic fields?
Ampere's law is most useful for situations with high symmetry, such as infinite wires or solenoids. For more complex geometries or finite structures, it may be difficult or impossible to choose an appropriate amperian loop, making the Biot-Savart law or numerical methods more suitable.
13. How does Ampere's law apply to a solenoid?
For an ideal, infinitely long solenoid, Ampere's law shows that the magnetic field inside is uniform and given by B = μ₀nI, where n is the number of turns per unit length and I is the current. Outside the solenoid, the magnetic field is theoretically zero.
14. How does Ampere's law apply to a toroidal coil?
For a toroidal coil, Ampere's law shows that the magnetic field is confined within the torus. The field strength inside the torus is B = μ₀NI / (2πr), where N is the total number of turns, I is the current, and r is the distance from the center of the torus to the point of measurement.
15. Can Ampere's law be used to calculate the magnetic field of a bar magnet?
Ampere's law in its basic form is not directly applicable to permanent magnets like bar magnets. It deals with currents, not magnetic materials. However, one can model a bar magnet as a collection of current loops (at the atomic level) and then apply Ampere's law to these equivalent currents.
16. How does Ampere's law relate to Gauss's law for magnetism?
While Ampere's law relates magnetic fields to electric currents, Gauss's law for magnetism states that the net magnetic flux through any closed surface is zero. Together, these laws form part of Maxwell's equations, providing a complete description of electromagnetic phenomena.
17. What is an amperian loop?
An amperian loop is an imaginary closed path used in applying Ampere's law. It can be any shape or size, but is often chosen to exploit symmetry in the problem to simplify calculations. The magnetic field is integrated around this loop to relate it to the enclosed current.
18. How does Ampere's law relate to the concept of curl in vector calculus?
Ampere's law is a physical manifestation of the mathematical concept of curl. The law states that the curl of the magnetic field (∇ × B) is proportional to the current density. This relationship is fundamental in understanding how currents generate magnetic fields.
19. What is the relationship between Ampere's law and Faraday's law of induction?
Ampere's law and Faraday's law are complementary: Ampere's law shows how currents create magnetic fields, while Faraday's law describes how changing magnetic fields induce currents. Together, they form part of the fundamental relationships in electromagnetism.
20. Can Ampere's law be used to calculate the magnetic field of a finite solenoid?
Ampere's law in its simple form is not directly applicable to finite solenoids due to the lack of symmetry at the ends. However, it can be used to approximate the field near the center of a long solenoid. For accurate calculations of finite solenoids, the Biot-Savart law or numerical methods are preferred.
21. How does Ampere's law apply to a sheet of current?
For an infinite sheet of current, Ampere's law shows that the magnetic field is constant on either side of the sheet and parallel to the sheet. The magnitude is B = μ₀K/2, where K is the surface current density. The field directions on opposite sides of the sheet are antiparallel.
22. What is the significance of the displacement current in the modified Ampere's law?
The displacement current, introduced by Maxwell, accounts for the magnetic field produced by changing electric fields, even in the absence of conduction current. This addition makes Ampere's law consistent with the law of conservation of charge and allows for the description of electromagnetic waves.
23. How does Ampere's law apply to a cylindrical shell carrying current?
For a cylindrical shell carrying current, Ampere's law shows that the magnetic field inside the shell is zero (as no current is enclosed), while outside it's the same as for a long straight wire carrying the same current. This demonstrates the importance of considering the current enclosed by the amperian loop.
24. Can Ampere's law be used to find the magnetic field of a moving charge?
Ampere's law in its basic form deals with steady currents, not individual moving charges. However, for a continuous stream of moving charges (which is effectively a current), Ampere's law can be applied. For a single moving charge, the Biot-Savart law is more appropriate.
25. How does Ampere's law relate to the concept of magnetic flux?
While Ampere's law doesn't directly involve magnetic flux, it's closely related. The law essentially states that the circulation of the magnetic field (which is related to the curl) around a closed loop is proportional to the current through the loop. This circulation is different from, but related to, the concept of magnetic flux through a surface.
26. What is the role of Ampere's law in understanding electromagnetic induction?
Ampere's law, especially in its modified form including displacement current, is crucial for understanding electromagnetic induction. It shows how changing electric fields can produce magnetic fields, which is the basis for the generation of electromagnetic waves and the operation of many electrical devices.
27. How does Ampere's law apply to a uniformly magnetized material?
For a uniformly magnetized material, Ampere's law can be applied by considering the equivalent surface current density due to the alignment of magnetic dipoles. Inside the material, the bound currents cancel, while at the surface, there's an effective current that can be used with Ampere's law to find the H-field.
28. What is the significance of Ampere's law in the development of electromagnetic theory?
Ampere's law was a crucial step in unifying electricity and magnetism. It provided a mathematical relationship between electric currents and magnetic fields, paving the way for Maxwell's equations and the complete theory of electromagnetism, which in turn led to the understanding of electromagnetic waves and modern communications technology.
29. How does Ampere's law apply to a Helmholtz coil configuration?
For a Helmholtz coil (two identical circular coils separated by a distance equal to their radius), Ampere's law can be applied to each coil individually. The total field is then found by superposition. This configuration is designed to produce a nearly uniform magnetic field in the central region between the coils.
30. Can Ampere's law be used to calculate the magnetic field inside a plasma?
Yes, Ampere's law can be applied to plasmas, which are essentially ionized gases carrying currents. However, the application can be complex due to the dynamic nature of plasmas. In many cases, the law is used in conjunction with other plasma physics equations to describe the magnetic fields and current distributions in plasmas.
31. How does Ampere's law relate to the concept of magnetic vector potential?
The magnetic vector potential (A) is related to the magnetic field (B) by B = ∇ × A. Ampere's law can be expressed in terms of A, leading to a differential equation for the vector potential. This formulation is particularly useful in more advanced treatments of electromagnetism and in quantum mechanics.
32. What is the importance of Ampere's law in the design of electromagnets?
Ampere's law is crucial in electromagnet design as it relates the current in the windings to the magnetic field produced. It helps in determining the number of turns and current required to achieve a desired field strength, and in understanding how the field varies within and around the electromagnet.
33. How does Ampere's law apply to a current-carrying loop?
For a circular current-carrying loop, Ampere's law can be used to find the magnetic field along the axis of the loop. Off the axis, the application becomes more complex, and often the Biot-Savart law is preferred. The law demonstrates how the field varies with distance from the loop's center.
34. Can Ampere's law be used to explain the Earth's magnetic field?
While Ampere's law is fundamental to understanding magnetic fields, it's not directly applicable to explaining the Earth's magnetic field. The Earth's field is generated by complex currents in its liquid outer core, a process better described by magnetohydrodynamics. However, Ampere's law is still crucial in modeling and understanding the overall behavior of planetary magnetic fields.
35. How does Ampere's law relate to the concept of magnetic dipoles?
Ampere's law can be used to understand magnetic dipoles by considering them as small current loops. The law shows how the magnetic field of a dipole varies with distance and direction, which is crucial in understanding the behavior of materials with magnetic dipole moments, such as permanent magnets.
36. What is the significance of Ampere's law in understanding transformers?
Ampere's law is fundamental in understanding transformer operation. It explains how changing current in the primary coil creates a changing magnetic field, which then induces current in the secondary coil. The law helps in calculating the magnetic field strength within the transformer core and in designing efficient transformers.
37. How does Ampere's law apply to superconductors?
In superconductors, Ampere's law is still valid, but its application becomes interesting due to the Meissner effect. Inside a superconductor, the magnetic field is zero, so any amperian loop entirely within the superconductor encloses no net current. Surface currents in the superconductor generate fields that exactly cancel external fields.
38. Can Ampere's law be used to explain the magnetic field of a rotating charged sphere?
Yes, Ampere's law can be applied to a rotating charged sphere. The rotation of charge is equivalent to a current distribution, and Ampere's law can be used to find the resulting magnetic field. This concept is important in understanding the magnetic fields of rotating celestial bodies.
39. How does Ampere's law relate to the concept of magnetic circuits?
Ampere's law is fundamental to the concept of magnetic circuits. Just as Kirchhoff's voltage law applies to electric circuits, Ampere's law applies to magnetic circuits. It relates the magnetomotive force (analogous to voltage) to the magnetic field strength and path length around the circuit.
40. What is the role of Ampere's law in understanding magnetic confinement fusion?
In magnetic confinement fusion, Ampere's law is crucial for understanding how the magnetic fields used to confine the plasma are generated and maintained. It helps in designing the coil systems that create the necessary magnetic field configurations to contain the high-temperature plasma.
41. How does Ampere's law apply to a system of parallel current-carrying wires?
For parallel current-carrying wires, Ampere's law can be used to find the magnetic field around each wire. The total field is then found by superposition. This application is important in understanding the forces between current-carrying conductors and in the design of electrical systems.
42. Can Ampere's law be used to explain the Hall effect?
While Ampere's law doesn't directly explain the Hall effect, it's crucial in understanding the magnetic field that causes it. The law helps in calculating the magnetic field strength in the Hall effect setup, which then interacts with moving charges to produce the Hall voltage.
43. How does Ampere's law relate to the concept of magnetic levitation?
Ampere's law is fundamental in understanding magnetic levitation. It explains how currents in superconductors or electromagnets generate the magnetic fields necessary for levitation. The law helps in calculating the field strengths required to counteract gravity and achieve stable levitation.
44. What is the significance of Ampere's law in the operation of electric motors?
Ampere's law is crucial in electric motor operation. It explains how current in the motor windings creates magnetic fields that interact with permanent magnets or other field windings. This interaction produces the torque that causes the motor to rotate. The law is used in motor design to optimize field strength and efficiency.
45. How does Ampere's law apply to a toroidal transformer?
In a toroidal transformer, Ampere's law shows that the magnetic field is confined within the toroidal core. The law helps in calculating the field strength inside the core, which is crucial for determining the transformer's performance. It also explains why toroidal transformers have less magnetic leakage than other designs.
46. Can Ampere's law be used to understand the magnetic fields in astrophysical jets?
Yes, Ampere's law is fundamental in understanding the magnetic fields associated with astrophysical jets. It helps explain how currents in the plasma of these jets generate and maintain the magnetic fields that collimate the jet material. The law is used in models of jet formation and propagation.

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