Home › Physics › Electricity and Magnetism › Applications of Ampere's Circuital Law
Electricity and Magnetism
Applications of Ampere's Circuital Law
Ampere's circuital law relates the line integral of the magnetic field B along a closed loop to the net algebraic current passing through the surface bounded by the loop. It serves as a powerful analytical tool for determining magnetic fields produced by highly symmetric current distributions.
UP TGT ScienceUP PGT PhysicsStudy Notes + Practice
Scope and prerequisites
TGT core
Calculation of magnetic field due to a long straight current-carrying wire, infinite sheet of current, and inside a long straight solenoid using Ampere's circuital law.
PGT extension
Magnetic field calculations for a toroidal solenoid, thick cylindrical conductors, coaxial cables, and boundary conditions for magnetic fields.
Prerequisites
Line integrals and closed path integration in vector calculus
Biot-Savart law and the principle of superposition
Concept of enclosed current and surface threading in magnetostatics
Statement of Ampere's Circuital Law
The line integral of the magnetic field vector B around any closed loop is equal to μ0 times the total steady current Ienclosed threading the surface.
Mathematically expressed as ∮ B · d$l = \mu_{0} I_{enclosed}$, where μ0 is the permeability of free space.
It is valid for steady currents and holds universally in magnetostatics, analogous to Gauss's law in electrostatics.
Symmetry Requirements for Application
The application of Ampere's law to evaluate B analytically requires high spatial symmetry (cylindrical, planar, or toroidal).
An Amperian loop must be chosen such that the magnitude of B is constant along the path and B is either parallel or perpendicular to the path element dl.
When B is perpendicular to dl, the dot product vanishes; when parallel and uniform, B factors out of the integral.
Magnetic Field of a Long Straight Solenoid
An ideal solenoid is treated as closely wound helical turns where the magnetic field outside is zero and uniform inside along the axis.
An rectangular Amperian loop is chosen with one side parallel to the axis inside and the opposite side far outside where B = 0.
The field magnitude depends strictly on the turn density n and current I, independent of the radius of the solenoid.
Magnetic Field of a Toroidal Solenoid
A toroid is a hollow circular ring clothed with closely spaced current-carrying turns, confining the magnetic field entirely within the core.
Due to axial symmetry, the Amperian loop is a circle of radius r concentric with the toroid axis.
The magnetic field drops to zero in the open space interior to the inner radius and exterior to the outer radius of the toroid.
Derivations
Magnetic Field Due to an Infinite Straight Current-Carrying Wire
Conditions: Steady current I in an infinitely long, thin straight wire, evaluated at a perpendicular distance r.
Establish cylindrical symmetry where magnetic field lines form concentric circles around the wire.
Choose a circular Amperian loop of radius r centered on the wire, lying in a plane perpendicular to it.
Evaluate the line integral ∮ B · d$l = B \oint dl = B(2\pir)$ since B is uniform in magnitude and parallel to dl everywhere on the loop.
Equate the line integral to μ0Ienclosed according to Ampere's circuital law: B(2πr) = μ0I.
Result: $B = \frac{\mu_0 I}{2\pi r}$
Magnetic Field Inside and Outside a Long Straight Solenoid
Conditions: Solenoid of length L with N total turns, turn density $n = N/L$, carrying current I.
Consider a rectangular Amperian loop of length h, with one side of length h inside the solenoid parallel to the axis, one side outside where B=0, and two radial sides perpendicular to the field lines.
Compute the line integral over the four segments: the integral vanishes along the two radial sides (B ⊥ dl) and the outside side (B = 0).
The total line integral reduces to the contribution along the inner side of length h: ∮ B · dl = Bh.
Calculate the enclosed current: the number of turns enclosed by the loop is nh, so Ienclosed = nhI.
Apply Ampere's law: B$h = \mu_{0}(nhI)$.
Result: $B = \mu_0 n I$
Formula sheet
Formula / law
Equation
Variables and units
Conditions
Ampere's Circuital Law
$$\oint B \cdot dl = \mu_0 I_{\text{enclosed}}$$
B = magnetic field, dl = elemental path vector, μ0 = permeability of free space, Ienclosed = net current passing through loop T m (Tesla meter)
Steady currents and static magnetic fields
Magnetic Field of Long Straight Wire
$$B = \frac{\mu_0 I}{2\pi r}$$
I = current, r = perpendicular distance from wire axis Tesla (T)
Infinite length approximation
Magnetic Field Inside Long Solenoid
$$B = \mu_0 n I$$
n = number of turns per unit length, I = current Tesla (T)
Long solenoid, evaluated far from ends
Magnetic Field of Toroidal Solenoid
$$B = \frac{\mu_0 N I}{2\pi r}$$
N = total number of turns, r = radial distance from toroid center Tesla (T)
Inside the core of the toroid (a < r < b)
Magnetic Field Inside Solid Cylindrical Conductor (r < R)
$$B = \frac{\mu_0 I r}{2\pi R^2}$$
I = total uniform current, R = radius of cylinder, r = radial distance from center Tesla (T)
Uniform current density across cross-section
Magnetic Field Due to Infinite Current Sheet
$$B = \frac{\mu_0 K}{2}$$
K = surface current density (current per unit width) Tesla (T)
Infinite flat sheet carrying uniform surface current
Differential Form of Ampere's Law
$$\nabla \times B = \mu_0 J$$
J = current density vector, ∇ $\times = curl operator$ T/m
Valid in continuous media for magnetostatics
Magnetic Field at the Ends of a Finite Solenoid
$$B = \frac{\mu_0 n I}{2}$$
n = turns per unit length, I = current Tesla (T)
2end of a very long solenoid
Graphs and experiments
B versus r for a Long Straight Conductor
Draw: A plot showing B on the y-axis and radial distance r on the x-axis. For r < R (inside), B increases linearly (B ∝ r). For r > R (outside), B drops off in a rectangular hyperbola (B ∝ 1/r).
Interpret: Illustrates how magnetic field behavior transitions from interior current distribution to exterior field dependence.
Exam use: Tested in multiple-choice questions asking to identify correct graphical variations of B with distance for solid conductors.
B versus r for a Toroidal Solenoid
Draw: A plot where B is zero for r < a (inner radius), follows a 1/r curve between a and b (core region), and drops abruptly to zero for r > b (outer radius).
Interpret: Confines magnetic field strictly to the core volume, demonstrating the ideal shielding property of toroids.
Exam use: Frequently used to test understanding of non-zero regions of magnetic field in closed geometries.
Mapping Magnetic Field of a Solenoid using Iron Filings or Compass
Draw: A schematic showing parallel field lines inside the solenoid looping around to form continuous closed paths outside.
Interpret: Demonstrates that magnetic field lines are continuous closed loops with no magnetic monopoles.
Exam use: Conceptual questions on directional properties and continuity of B-fields.
Solved numericals
TGT core
A long straight wire carries a current of 10 A. Calculate the magnitude of the magnetic field at a perpendicular 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.
State the formula for a straight wire: $B = (\mu_{0}I) / (2\pir)$.
A solenoid is 0.5 m long, has 500 turns of wire, and carries a current of 2 A. Find the magnitude of the magnetic field inside the solenoid near its center.
Calculate turn density $n = N / L = 500 / 0.5 = 1000 turns/m$.
A toroidal solenoid with a mean radius of 0.2 m has 1000 turns wound uniformly. If it carries a current of 0.5 A, find the magnetic field at a point inside the core at distance 0.2 m from the center.
Identify given parameters: N = 1000, r = 0.2 m, I = 0.5 A.
State formula for toroid field: $B = (\mu_{0} N I) / (2\pir)$.
Ampere's circuital law in its standard integral form is strictly valid only for steady currents (magnetostatics) where electric fields do not change with time.
For time-varying electromagnetic fields, Maxwell's displacement current correction must be added to the right side of the law.
Common misconceptions
Common error: Ampere's circuital law can only be used to calculate magnetic fields when there is high symmetry.
Correct: Ampere's law is always true for any closed loop and any current distribution, but it is only useful for calculating B analytically when high symmetry allows B to be factored out of the integral.
The law is a universal topological relation, but algebraic evaluation requires symmetry.
Common error: The enclosed current in Ampere's law includes currents situated outside the Amperian loop.
Correct: Only currents that pierce through the open surface bounded by the Amperian loop contribute to Ienclosed.
External currents produce magnetic fields that contribute to B at every point on the path, but their net circulation around the closed loop sums to zero.
Quick revision
Ampere's law states ∮ B · d$l = \mu_{0}I_{enclosed}$.
Straight wire field: $B = \mu_{0}I / (2\pir)$.
Solenoid field: $B = \mu_{0}nI$.
Toroid field: $B = \mu_{0}NI / (2\pir)$.
Applicable only for steady currents unless Maxwell's displacement current is included.
Must remember
Ampere's Circuital Law states that the line integral of $\mathbf{B} \cdot d\mathbf{l}$ around any closed loop equals $\mu_0 I_{\text{enclosed}}$.
The magnetic field inside an ideal long solenoid of turn density $n$ and current $I$ is given by $B = \mu_0 n I$.
For a toroidal solenoid with total turns $N$ and radius $r$, the magnetic field inside the core is $B = \frac{\mu_0 N I}{2\pi r}$.
Ampere's law is always universally valid, but it can be used for analytical calculation of $\mathbf{B}$ only when high spatial symmetry exists.
The magnetic field inside a solid cylindrical conductor of radius $R$ carrying uniform current $I$ at a distance $r < R$ is $B = \frac{\mu_0 I r}{2\pi R^2}$.
Glossary
Amperian LoopA closed auxiliary mathematical path chosen to evaluate the line integral of the magnetic field.
Permeability of Free Space ($\mu_0$)A physical constant representing the magnetic penetrability of vacuum, equal to $4\pi \times 10^{-7} \text{ T m A}^{-1}$.
Surface Current Density ($K$)Current per unit width flowing across a sheet, measured in amperes per meter (A/m).
ToroidA donut-shaped coil of insulated wire wound on a circular ring form, used to confine magnetic fields.
Solenoid vs. Toroid Magnetic Fields
Feature
Long Straight Solenoid
Toroidal Solenoid
Field Geometry
Uniform inside, parallel to axis
Concentric circular field lines inside core
Field Dependence on Radius ($r$)
Independent of radial distance $r$
Inversely proportional to radius ($B \propto 1/r$)
External Field
Zero outside the solenoid
Zero in both inner and outer empty spaces
Practice Quiz
Choose an option to reveal the explanation.
1. What does the line integral $\oint \mathbf{B} \cdot d\mathbf{l}$ around a closed loop evaluate to according to Ampere's Circuital Law?
Explanation: Ampere's Circuital Law mathematically states that the line integral of the magnetic field around any closed loop is equal to $\mu_0$ times the total steady current enclosed by the loop.
2. What is the magnetic field inside a long straight solenoid having $n$ turns per unit length carrying current $I$?
Explanation: Using an Amperian rectangular loop, the magnetic field inside a long ideal solenoid is evaluated as $B = \mu_0 n I$.
3. The magnetic field inside a toroidal solenoid at a radial distance $r$ from its center with $N$ total turns and current $I$ is given by:
Explanation: Due to axial symmetry, applying Ampere's circuital law along a circular path of radius $r$ inside a toroid yields $B = \frac{\mu_0 N I}{2\pi r}$.
4. What is the magnetic field at the ends of a very long solenoid?
Explanation: At the ends of a very long solenoid, the magnetic field drops to half of its interior value, becoming $B = \frac{\mu_0 n I}{2}$.
5. Which physical law in electrostatics is mathematically analogous to Ampere's Circuital Law in magnetostatics?
Explanation: Ampere's law for magnetic fields is analogous to Gauss's law for electric fields, as both relate a field line integral over a closed boundary to the source enclosed.
6. What is the magnetic field at a point situated at a radial distance $r$ inside ($r < R$) a solid cylindrical conductor of radius $R$ carrying a total uniform current $I$?
Explanation: By applying Ampere's law, the enclosed current for $r < R$ is $I_{\text{enclosed}} = I (r^2 / R^2)$, leading to $B = \frac{\mu_0 I r}{2\pi R^2}$.
7. What is the differential form of Ampere's Circuital Law in magnetostatics?
Explanation: The differential form of Ampere's Circuital Law is $\nabla \times \mathbf{B} = \mu_0 \mathbf{J}$, where $\mathbf{J}$ is the current density vector.
8. What is the magnetic field due to an infinite current sheet carrying a surface current density $K$?
Explanation: Using a rectangular Amperian loop straddling the infinite current sheet, the magnetic field is found to be uniform on either side with magnitude $B = \frac{\mu_0 K}{2}$.
9. Why is Ampere's Circuital Law not useful for calculating the magnetic field of a single finite current-carrying loop of arbitrary shape?
Explanation: Analytical evaluation using Ampere's law strictly requires high spatial symmetry (such as cylindrical, planar, or toroidal) so that $B$ can be factored out of the path integral.
10. In a toroidal solenoid, what is the magnetic field in the empty open space interior to the inner radius?
Explanation: An Amperian loop drawn in the interior empty space of a toroid encloses zero net current, meaning the magnetic field drops to zero.
11. When can the magnetic field vector $\mathbf{B}$ be factored directly out of the Ampere's law line integral $\oint \mathbf{B} \cdot d\mathbf{l}$?
Explanation: If the path has symmetry such that $B$ is uniform in magnitude and parallel to $d\mathbf{l}$, the dot product becomes $B \, dl$ and $B$ factors out.
12. Which currents contribute to $I_{\text{enclosed}}$ when evaluating Ampere's Circuital Law for a given closed Amperian loop?
Explanation: Only the net current threading (piercing) the surface bounded by the closed Amperian loop contributes to $I_{\text{enclosed}}$.
13. What is the SI unit of the magnetic field $B$?
Explanation: The SI unit of magnetic field is the Tesla (T), which is equivalent to a Weber per square meter.
14. What is the magnetic field at a radial distance $r$ outside ($r > R$) a long solid cylindrical conductor of radius $R$ carrying total current $I$?
Explanation: For an Amperian loop outside the solid cylinder ($r > R$), the total enclosed current is simply $I$, yielding $B = \frac{\mu_0 I}{2\pi r}$, identical to a thin wire.
15. If a closed Amperian loop encloses two equal and opposite currents threading through it, what is the net enclosed current $I_{\text{enclosed}}$?
Explanation: Equal and opposite currents cancel each other out algebraically when finding net current, resulting in $I_{\text{enclosed}} = 0$.
PYQ Patterns
These are pattern-based practice questions. Authentic year labels are only used when verified from a source.
1. A long solenoid of length $L$ has $N$ total turns and carries a current $I$. What is the magnetic field inside the solenoid near its center?
Explanation: The turn density $n = N/L$. Substituting into the formula $B = \mu_0 n I$ gives $B = \frac{\mu_0 N I}{L}$.
2. A toroid has an inner radius $a$ and outer radius $b$, with a total of $N$ turns carrying current $I$. The magnetic field at a radial distance $r$ ($a < r < b$) inside the core is:
Explanation: By Ampere's law, for any point inside the core of a toroid at distance $r$, the magnetic field is $B = \frac{\mu_0 N I}{2\pi r}$.
3. A long straight wire of radius $R$ carries a steady current $I$ distributed uniformly across its cross-section. The ratio of magnetic fields at $r = R/2$ and $r = 2R$ is:
Explanation: At $r = R/2$ (inside), $B_{\text{in}} = \frac{\mu_0 I (R/2)}{2\pi R^2} = \frac{\mu_0 I}{4\pi R}$. At $r = 2R$ (outside), $B_{\text{out}} = \frac{\mu_0 I}{2\pi(2R)} = \frac{\mu_0 I}{4\pi R}$. Thus, the ratio is $1:1$.
4. Which of the following statements regarding Ampere's Circuital Law is correct?
Explanation: Ampere's Circuital Law states that $\oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{\text{enclosed}}$, meaning the circulation depends strictly on the net current passing through the surface bounded by the loop.
5. A flat infinite sheet carries a uniform surface current density $K$ along the y-axis. The magnitude of the magnetic field at a small distance $z$ from the sheet is:
Explanation: The magnetic field due to an infinite current sheet is $B = \frac{\mu_0 K}{2}$, which is uniform and independent of the perpendicular distance $z$.
6. In a coaxial cable, the inner conductor carries current $I$ outwards and the outer cylindrical shell carries current $I$ inwards. What is the magnetic field in the region outside the outer conductor?
Explanation: Outside the coaxial cable, an Amperian loop encloses a total net current of $I_{\text{enclosed}} = I - I = 0$. Hence, the magnetic field is zero.
Timed Topic Test
Attempt these questions under a ten-minute limit using a separate timer.
1. What is the primary condition required to evaluate the magnetic field analytically using Ampere's Circuital Law?
Explanation: High spatial symmetry allows the magnetic field magnitude to be constant along the Amperian loop and parallel to $d\mathbf{l}$, enabling it to be factored out of the integral.
2. A solenoid of length $0.5\text{ m}$ has 1000 turns and carries a current of $2\text{ A}$. What is the magnetic field at its center? (Take $\mu_0 = 4\pi \times 10^{-7}\text{ T m A}^{-1}$)
Explanation: First find $n = N/L = 1000 / 0.5 = 2000\text{ turns/m}$. Then $B = \mu_0 n I = (4\pi \times 10^{-7})(2000)(2) = 1.6\pi \times 10^{-3} = 5.02\times 10^{-3}\text{ T}$ (approx $5\pi \times 10^{-3}\text{ T}$).
3. If the current in a long straight solenoid is doubled while keeping its turn density constant, what happens to the magnetic field inside?
Explanation: Since $B = \mu_0 n I$, the magnetic field $B$ is directly proportional to the current $I$. Doubling the current doubles the magnetic field.
4. What is the value of the line integral $\oint \mathbf{B} \cdot d\mathbf{l}$ around a closed path if a long straight wire carrying current $I$ passes *outside* the closed path?
Explanation: If the current-carrying wire is outside the Amperian loop, the net current enclosed by the loop is zero, making the line integral zero.
5. A toroid has 500 turns and a mean radius of $0.1\text{ m}$, carrying a current of $1\text{ A}$. What is the magnetic field inside the core?
6. Which of the following statements is a common misconception about Ampere's Circuital Law?
Explanation: A common misconception is that external currents contribute to $I_{\text{enclosed}}$. Only currents piercing through the open surface bounded by the Amperian loop are included.
7. In a thick cylindrical wire of radius $R$, the magnetic field at the surface ($r = R$) is $B_0$. What is the magnetic field at $r = R/2$?
Explanation: At the surface ($r = R$), $B_0 = \frac{\mu_0 I}{2\pi R}$. At $r = R/2$, $B = \frac{\mu_0 I (R/2)}{2\pi R^2} = \frac{\mu_0 I}{4\pi R} = \frac{B_0}{2}$.
8. What is the nature of the magnetic field outside an ideal long straight solenoid?
Explanation: For an ideal long solenoid, the magnetic field outside is zero because the field lines form closed loops entirely confined within the solenoid core.
9. Which variable does NOT affect the magnetic field inside an ideal infinite solenoid?
Explanation: The formula $B = \mu_0 n I$ shows that the magnetic field of an ideal solenoid is independent of its radius.
10. What does the curl operator acting on the magnetic field vector in magnetostatics ($\nabla \times \mathbf{B}$) physically represent?
Explanation: According to the differential form of Ampere's law, $\nabla \times \mathbf{B} = \mu_0 \mathbf{J}$, the curl of $\mathbf{B}$ is proportional to the local current density vector $\mathbf{J}$.