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Electricity and Magnetism

Ampere's Circuital Law

Ampere's circuital law relates the line integral of a magnetic field around a closed loop to the total electric current passing through the surface bounded by the loop. It serves as a fundamental principle in magnetostatics, analogous to Gauss's law in electrostatics.

UP TGT ScienceUP PGT PhysicsStudy Notes + Practice

Scope and prerequisites

TGT core

Direct application of Ampere's law to find magnetic fields due to symmetric current distributions such as long straight wires, infinite sheets, and long solenoids.

PGT extension

Advanced calculations involving toroids, non-uniform current densities, magnetization vectors, verification of Stokes' theorem, and the inclusion of Maxwell's displacement current term.

Prerequisites

Biot-Savart law and magnetic field due to steady currents
Line integrals and closed-loop integration in vector calculus
Right-hand thumb rule for the direction of magnetic fields

Statement and Mathematical Formulation

Ampere's circuital law states that the line integral of the magnetic field B over a closed path is equal to μ₀ times the net steady current I_enclosed threading through the surface bounded by the path.
Mathematically expressed as ∮ B · d$l = \mu_{0} I_{enclosed} in vacuum or free space$.
The law is universally valid in magnetostatics, but fails for time-varying electric fields unless Maxwell's displacement current is added.

Choice of Amperian Loop

An Amperian loop is a hypothetical closed curve chosen for evaluating the line integral of the magnetic field.
To simplify the calculation of B · dl, the loop must possess high symmetry such that the magnitude of B is uniform along the path and B is either parallel or perpendicular to the path element dl.
The shape of the Amperian loop does not affect the value of the line integral, provided the enclosed current remains unchanged.

Sign Convention for Enclosed Current

The direction of integration along the Amperian loop defines a positive sense via the right-hand screw rule.
Currents piercing the surface in the direction of the right-hand thumb curl are taken as positive.
Currents piercing in the opposite direction are taken as negative when computing the net enclosed current.

Differential Form of Ampere's Law

Using Stokes' theorem, the line integral of B over a closed loop can be converted into a surface integral of the curl of B.
This yields ∇ × $B = \mu_{0} J$, where J is the free current density vector.
The differential form relates the local properties of the magnetic field directly to the local current density at any point.

Derivations

Magnetic Field of an Infinite Long Straight Conductor

Conditions: Steady current I flowing through an infinitely long, thin straight wire, evaluated at a radial distance r.

By symmetry, the magnetic field lines are concentric circles centered on the wire, with B having a constant magnitude at a fixed radius r and pointing tangentially.
Choose a circular Amperian loop of radius r centered on the wire, coaxial with the conductor.
Evaluate the line integral: ∮ B · d$l = \oint B dl \cos(0°) = B \oint dl = B (2\pir)$.
Equate this to μ₀ I_enclosed according to Ampere's law, giving B (2πr) = μ₀ I.

Result: $B = \frac{\mu_0 I}{2\pi r}$

Magnetic Field inside and outside a Long Solenoid

Conditions: Solenoid of n turns per unit length carrying current I, assuming an infinitely long solenoid for edge-effect neglect.

Argue that the magnetic field outside an ideal infinite solenoid is zero, and inside it is uniform and parallel to the axis.
Choose a rectangular Amperian loop of length l, with one side parallel to the axis inside the solenoid, the opposite side outside, and two sides perpendicular to the axis.
Evaluate the line integral along the four segments: only the segment inside the solenoid contributes to the integral, yielding B × l.
The total current enclosed by this rectangular loop is the number of turns inside length l multiplied by current I, which is (nl)I.

Result: $B = \mu_0 n I$

Formula sheet

Formula / law	Equation	Variables and units	Conditions
Integral form of Ampere's Law	$$\oint B \cdot dl = \mu_0 I_{\text{enclosed}}$$	B = magnetic field, dl = elemental path length, μ₀ = permeability of free space, I_enclosed = net current T · m (Tesla-meter)	Steady currents in free space or vacuum
Differential form of Ampere's Law	$$\nabla \times B = \mu_0 J$$	∇ × B = curl of magnetic field, J = current density T/m or A/m²	Magnetostatic field conditions
Magnetic field of a long straight wire	$$B = \frac{\mu_0 I}{2\pi r}$$	I = current, r = perpendicular distance from wire Tesla (T)	Infinitely long straight conductor
Magnetic field inside an ideal solenoid	$$B = \mu_0 n I$$	n = turns per unit length, I = current Tesla (T)	Long uniform solenoid, interior point
Magnetic field inside a Toroid	$$B = \frac{\mu_0 N I}{2\pi r}$$	N = total number of turns, r = radial distance from toroid center Tesla (T)	Inside the core of a toroid of radius r
Magnetic field near an infinite sheet of current	$$B = \frac{\mu_0 K}{2}$$	K = surface current density (current per unit width) Tesla (T)	Infinite sheet carrying uniform surface current
Ampere-Maxwell Law (with displacement current)	$$\oint B \cdot dl = \mu_0 \left( I_{\text{enclosed}} + \epsilon_0 \frac{d\Phi_E}{dt} \right)$$	ε₀ = permittivity of free space, Φ_E = electric flux T · m	Time-varying electric fields present
Magnetic field inside a thick cylindrical conductor (r < R)	$$B = \frac{\mu_0 I r}{2\pi R^2}$$	R = radius of cylinder, r = distance from axis Tesla (T)	Uniform current distribution across cross-section

Graphs and experiments

B versus r for a Long Straight Conductor

Draw: A graph with radial distance r on the x-axis and magnetic field B on the y-axis. Inside the wire (r < R), B increases linearly with r. Outside the wire (r > R), B decreases hyperbolically as 1/r.

Interpret: Shows the transition of magnetic field behavior from inside a solid current-carrying conductor to the external space.

Exam use: Frequently tested in multiple choice questions regarding matching graphs to conductor geometries.

B versus r for a Toroidal Coil

Draw: A plot showing zero magnetic field for r less than the inner radius, a 1/r decay profile within the core region, and zero field again for r greater than the outer radius.

Interpret: Demonstrates the containment of the magnetic field entirely within the toroidal core.

Exam use: Used to test understanding of boundary conditions and zero flux leakage in closed magnetic circuits.

Rowland Ring Experiment

Draw: A schematic or conceptual graph illustrating the measurement of magnetic induction B inside a toroidal specimen as a function of magnetizing field H.

Interpret: Validates Ampere's circuital law applied to magnetic materials and helps determine relative permeability.

Exam use: Theoretical questions on experimental setup for measuring magnetic properties of materials.

Solved numericals

TGT core

A long straight wire carries a current of 10 A. Calculate the magnitude of the magnetic field at a distance of 5 cm from the wire.

Identify given values: I = 10 A, r = 5 cm = 0.05 m, μ₀ = 4π × 10^-7 T·m/A.
Apply the formula for a straight wire: $B = (\mu_{0} I) / (2\pir)$.
Substitute the values: $B = (4\pi \times 10^{-7} \times 10) / (2\pi \times 0.05)$.
Simplify the expression: $B = (2 \times 10^{-6}) / (0.05) = 4 \times 10^{-5} T$.

Answer: 4 × 10^-5 T

TGT core

A solenoid 50 cm long has 4 layers of windings of 400 turns each and carries a current of 2 A. Find the magnetic field at the center of the solenoid.

Calculate total turns: $N = 4 \times 400 = 1600 turns$.
Calculate turns per unit length n: $n = N / L = 1600 / 0.5 m = 3200 turns/m$.
Use the solenoid formula: $B = \mu_{0} n I$.
Substitute values: $B = (4\pi \times 10^{-7}) \times 3200 \times 2 = 2.56\pi \times 10^{-3} T \approx 8.04 \times 10^{-3} T$.

Answer: 8.04 × 10^-3 T

PGT extension

A long solid cylindrical wire of radius $R = 2 cm carries a total current I = 5 A distributed uniformly across its cross-section$. Calculate the magnetic field at a point r = 1 cm from the axis inside the wire.

Since the current is uniformly distributed, the current enclosed within radius r (1 cm) is proportional to the area ratio: I_enclosed = I × (r² / R²).
Substitute the values: I_enclosed = 5 × (1² / 2²) = 5 / 4 = 1.25 A.
Apply Ampere's law for inside a cylinder: B × (2πr) = μ₀ I_enclosed.
Rearrange for B: $B = (\mu_{0} I_{enclosed}) / (2\pir) = (4\pi \times 10^{-7} \times 1.25) / (2\pi \times 0.01)$.
Calculate final numerical value: $B = (2 \times 10^{-7} \times 1.25) / 0.01 = 2.5 \times 10^{-5} T$.

Answer: 2.5 × 10^-5 T

Assumptions and limitations

Ampere's circuital law in its standard form assumes steady-state currents (magnetostatics) where charge does not accumulate over time.
It fails when applied to circuits with changing electric fields unless the Maxwell displacement current term is included.
While the law holds true for any arbitrary closed loop, it is only analytically useful for highly symmetric current configurations.

Common misconceptions

Common error: The magnetic field B in Ampere's law is produced only by the current enclosed by the Amperian loop.

Correct: The magnetic field B at any point is the resultant field produced by ALL currents in the system, whereas the line integral of B depends solely on the enclosed current.

The line integral isolates the enclosed current due to vector cancellation properties, but the field vector itself obeys superposition from all sources.

Common error: If the net current enclosed by an Amperian loop is zero, the magnetic field B must be zero everywhere along the loop.

Correct: A zero net enclosed current only guarantees that the line integral of B is zero, not that the magnetic field is zero at every point on the path.

Individual segments of the path can experience non-zero magnetic fields that cancel out algebraically over the complete closed loop.

Common error: Ampere's law can only be applied to circular loops.

Correct: An Amperian loop can be of any arbitrary geometric shape, though symmetric shapes are chosen for ease of calculation.

The validity of the vector line integral theorem is independent of the contour shape, provided the surface bounded by it is continuous.

Quick revision

Ampere's law equates ∮ B · dl to μ₀ I_enclosed.
For a long straight wire, $B = \mu_{0}I / 2\pir$.
Inside an ideal infinite solenoid, $B = \mu_{0}nI$.
Inside a toroid, $B = \mu_{0}NI / 2\pir$.
Addition of displacement current (ε₀ dΦ_E/dt) completes Ampere's law for time-varying fields.

Must remember

Ampere's circuital law states that the line integral of the magnetic field $B$ over a closed path equals $\mu_0 I_{\text{enclosed}}$.
The law is universally valid in magnetostatics but fails for time-varying electric fields unless Maxwell's displacement current is added.
The magnetic field $B$ at any point is produced by all currents in the system, whereas the line integral depends solely on the enclosed current.
For a long straight wire carrying current $I$, the magnetic field at distance $r$ is given by $B = \frac{\mu_0 I}{2\pi r}$.
The magnetic field inside an ideal solenoid is $B = \mu_0 n I$, where $n$ is the number of turns per unit length.
The differential form of Ampere's law is given by $\nabla \times B = \mu_0 J$, connecting curl of $B$ directly to current density $J$.

Glossary

Amperian LoopA hypothetical closed curve chosen for evaluating the line integral of the magnetic field using Ampere's law.

Permeability of Free SpaceThe constant $\mu_0$ representing the magnetic permeability of vacuum, equal to $4\pi \times 10^{-7} \, \text{T}\cdot\text{m}/\text{A}$.

Displacement CurrentA term introduced by Maxwell, $\epsilon_0 \frac{d\Phi_E}{dt}$, representing the equivalent current due to changing electric fields.

Surface Current DensityCurrent per unit width denoted by $K$, used for calculating magnetic fields near infinite sheets of current.

Solenoid vs. Toroid Magnetic Field

Parameter	Long Solenoid	Toroid
Magnetic Field Formula	$B = \mu_0 n I$	$B = \frac{\mu_0 N I}{2\pi r}$
Field Uniformity	Uniform inside, zero outside for infinite length	Varies inversely with radial distance $r$ inside core
Geometry	Linear straight cylinder	Circular ring / doughnut shape

Practice Quiz

Choose an option to reveal the explanation.

1. What is the mathematical expression for the integral form of Ampere's circuital law in vacuum?

Explanation: Ampere's circuital law states that the line integral of $B$ over a closed path is $\mu_0$ times the net steady current enclosed.

2. Which of the following conditions causes Ampere's circuital law in its original magnetostatic form to fail?

Explanation: Original Ampere's law fails for time-varying electric fields because curl of $B$ does not account for changing electric flux, requiring Maxwell's displacement current.

3. What is the magnitude of the magnetic field at a perpendicular distance $r$ from an infinitely long straight wire carrying current $I$?

Explanation: Applying Ampere's law to a circular loop of radius $r$ around a straight wire yields $\oint B \cdot dl = B(2\pi r) = \mu_0 I$, hence $B = \frac{\mu_0 I}{2\pi r}$.

4. What is the magnetic field inside an ideal long solenoid with $n$ turns per unit length carrying current $I$?

Explanation: Using a rectangular Amperian loop partly inside and partly outside the solenoid, the magnetic field inside is uniform and given by $B = \mu_0 n I$.

5. What is the differential form of Ampere's circuital law in magnetostatics?

Explanation: Using Stokes' theorem on the integral form, the differential form is derived as $\nabla \times B = \mu_0 J$, where $J$ is the free current density.

6. If the net current enclosed by an Amperian loop is zero, what can be concluded about the magnetic field along the loop?

Explanation: A zero net enclosed current only guarantees that $\oint B \cdot dl = 0$, not that $B$ is zero at every point on the path.

7. What is the magnetic field inside the core of a toroid of radius $r$ with a total of $N$ turns carrying current $I$?

Explanation: Applying Ampere's law along a circular path of radius $r$ inside the toroid gives $B(2\pi r) = \mu_0 N I$, leading to $B = \frac{\mu_0 N I}{2\pi r}$.

8. What is the magnetic field near an infinite sheet carrying a uniform surface current density $K$?

Explanation: Using a rectangular Amperian loop straddling the sheet, $\oint B \cdot dl = 2Bl = \mu_0 Kl$, which simplifies to $B = \frac{\mu_0 K}{2}$.

9. What is the modified Ampere-Maxwell law including displacement current?

Explanation: Maxwell added the displacement current term $\epsilon_0 \frac{d\Phi_E}{dt}$ to maintain consistency in non-steady electromagnetic fields.

10. What is the magnetic field inside a thick cylindrical conductor of radius $R$ carrying a uniform total current $I$ at a distance $r < R$ from the axis?

Explanation: Enclosed current for $r < R$ is $I_{\text{enclosed}} = I \left(\frac{r^2}{R^2}\right)$. Applying Ampere's law gives $B(2\pi r) = \mu_0 I \frac{r^2}{R^2}$, yielding $B = \frac{\mu_0 I r}{2\pi R^2}$.

11. Which theorem is used to convert the line integral of $B$ into a surface integral of the curl of $B$?

Explanation: Stokes' theorem relates the surface integral of the curl of a vector field over an open surface to its line integral around the bounding loop.

12. How is the positive sense of the enclosed current determined when applying Ampere's circuital law?

Explanation: The direction of integration along the Amperian loop defines a positive sense via the right-hand screw rule for piercing currents.

13. What is the unit of the line integral of the magnetic field $\oint B \cdot dl$?

Explanation: Since $B$ is in Tesla and $dl$ is in meters, the product integral unit is Tesla-meter ($\text{T}\cdot\text{m}$).

14. Which of the following is true regarding the choice of an Amperian loop for calculating magnetic fields?

Explanation: To simplify the calculation of $B \cdot dl$, the loop must possess symmetry such that $B$ magnitude is uniform and aligned with $dl$.

15. What produces the magnetic field vector $B$ measured at a specific point in space?

Explanation: The magnetic field at any point obeys the superposition principle and is produced by all currents in the system.

PYQ Patterns

These are pattern-based practice questions. Authentic year labels are only used when verified from a source.

1. A long straight wire carries a current $I$. What is the magnetic field at a distance $r$ from the wire?

Explanation: Direct application of Ampere's circuital law for a long straight conductor yields $B = \frac{\mu_0 I}{2\pi r}$.

2. The differential form of Ampere's circuital law for magnetostatics is expressed as:

Explanation: The differential relation derived using Stokes' theorem is $\nabla \times B = \mu_0 J$.

3. What is the magnetic field inside an ideal solenoid having $n$ turns per unit length and carrying a current $I$?

Explanation: Ampere's law on a rectangular loop shows that the magnetic field inside a long uniform solenoid is $B = \mu_0 n I$.

4. What modification did James Clerk Maxwell make to Ampere's circuital law to make it universally valid?

Explanation: Maxwell added the displacement current term to account for time-varying electric fields.

5. In a toroid with total $N$ turns and core radius $r$ carrying current $I$, the magnetic field inside the core is:

Explanation: Applying Ampere's law along a circular Amperian loop of radius $r$ inside the toroid gives $B = \frac{\mu_0 N I}{2\pi r}$.

6. If the net current enclosed by an Amperian loop is zero, which of the following statements is strictly correct?

Explanation: A zero net enclosed current implies $\oint B \cdot dl = 0$ via Ampere's law, but $B$ itself may be non-zero at individual points.

Timed Topic Test

Attempt these questions under a ten-minute limit using a separate timer.

1. What is the SI unit of magnetic field $B$?

Explanation: The SI unit of magnetic field strength $B$ is the Tesla (T).

2. For a thick cylindrical conductor of radius $R$ carrying uniform current $I$, what is the magnetic field at a point outside the cylinder ($r > R$) at distance $r$ from the axis?

Explanation: For points outside ($r > R$), the entire current $I$ is enclosed, so $B(2\pi r) = \mu_0 I \implies B = \frac{\mu_0 I}{2\pi r}$.

3. Which law or theorem allows the transformation of the closed line integral in Ampere's law into a surface integral?

Explanation: Stokes' theorem equates the line integral of a vector field around a closed loop to the surface integral of its curl.

4. What is the value of permeability of free space $\mu_0$?

Explanation: The permeability of free space $\mu_0$ has an exact defined value of $4\pi \times 10^{-7} \, \text{T}\cdot\text{m/A}$.

5. What is the magnetic field outside an ideal infinite solenoid?

Explanation: For an ideal infinitely long solenoid, the magnetic field outside is zero because field lines form closed loops confined entirely within the core.

6. How does the magnetic field inside a toroid vary with radial distance $r$ from its center?

Explanation: Inside a toroid, $B = \frac{\mu_0 N I}{2\pi r}$, showing that $B$ varies inversely with radial distance $r$.

7. If a closed Amperian loop encloses two currents of $3\text{A}$ in the positive direction and $1\text{A}$ in the opposite direction, what is the net enclosed current?

Explanation: Net current $I_{\text{enclosed}} = 3\text{A} - 1\text{A} = 2\text{A}$ taking opposing directions into account.

8. What is the dimension of the line integral of magnetic field $\oint B \cdot dl$?

Explanation: Since $\oint B \cdot dl = \mu_0 I_{\text{enclosed}}$, its dimension is $[\mu_0][I] = [M L T^{-2} A^{-2}][A] = [M L T^{-2} A^{-1}]$. (Alternatively, $B \cdot dl$ has units of Tesla-meter which evaluates to force per unit current).

9. Which of the following statements about Ampere's circuital law is false?

Explanation: Ampere's law is only useful and easy to apply when high symmetry exists; for arbitrary distributions, Biot-Savart law is generally required.

10. What represents the displacement current term in the Ampere-Maxwell law?

Explanation: The displacement current introduced by Maxwell is given mathematically by $\epsilon_0 \frac{d\Phi_E}{dt}$.