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Electricity and Magnetism
Biot-Savart Law and Magnetic Field Due to Steady Currents
The Biot-Savart law determines the magnetic flux density produced at a point by a steady current-carrying conductor element. It serves as the foundational fundamental law in magnetostatics, analogous to Coulomb's law in electrostatics.
UP TGT ScienceUP PGT PhysicsStudy Notes + Practice
Scope and prerequisites
TGT core
Differential and vector forms of Biot-Savart law, application to calculate magnetic field at the center of a circular current-carrying coil, and long straight straight conductor.
PGT extension
Magnetic field at an arbitrary point along the axis of a circular loop, magnetic field due to a finite straight wire, Helmholtz coils, and applications involving solenoids and toroids.
Prerequisites
Vector calculus (cross product and line integrals)
Steady electric current and current density
Principle of superposition for magnetic fields
Vector Form of Biot-Savart Law
The magnetic field contribution dB due to a current element Idl is directly proportional to the current I, the length of the element dl, the sine of the angle theta between dl and the position vector r, and inversely proportional to the square of the distance r.
Mathematically expressed as d$B = (\mu_0 / 4pi) * (I dl x r-hat) / r^2$, where mu_0 is the permeability of free space.
The direction of dB is perpendicular to the plane containing the current element dl and the position vector r, determined strictly by the right-hand screw rule.
Permeability of Free Space and Constants
The constant of proportionality is expressed as mu_0 / 4pi, with the exact SI value of mu_0 equal to 4pi * 10^-7 Tesla meter per Ampere (T m A^-1).
The unit of magnetic field B is the Tesla (T) or Weber per meter squared (Wb m^-2), and the CGS unit is the Gauss.
The law strictly applies to steady currents (magnetostatics) where charge density does not vary with time.
Principle of Superposition in Magnetism
The total magnetic field B at any point in space due to a complete current-carrying circuit is the vector sum of the magnetic field contributions dB from all individual current elements.
Expressed mathematically as the line integral B = integral dB over the entire closed loop.
Allows calculation of complex geometries by breaking them down into symmetric differential elements.
Comparison with Coulomb's Law
Both laws are inverse-square laws dependent on 1/r^2 distance variations.
Coulomb's law deals with scalar charges and produces fields directed along the line joining charges, whereas Biot-Savart law deals with vector current elements and produces fields perpendicular to the plane of the element and position vector.
Electrostatic field is conservative (curl E = 0), while magnetostatic field is solenoidal (divergence B = 0).
Derivations
Magnetic Field at the Center of a Circular Current-Carrying Loop
Conditions: Circular loop of radius R carrying steady current I, evaluated at the geometric center.
Consider a small current element Idl on a circular loop of radius R.
The angle theta between the current element dl and the position vector r (which is equal to R) is exactly 90 degrees for every point on the circular loop.
Integrate around the entire circumference of the circle: $B = integral dB = (\mu_0 I / 4pi R^2) * integral dl$, where integral dl = 2pi R.
Simplify the resulting expression to obtain the final formula.
Result: $B = (\mu_0 I) / (2R)$
Magnetic Field on the Axis of a Circular Current Loop
Conditions: Circular loop of radius R, current I, evaluated at an axial distance x from the center.
Consider a current element Idl at the bottom of the loop. The distance r from dl to the axial point P is $r = \sqrt(R^2 + x^2)$.
The angle between dl and r is 90 degrees, so d$B = (\mu_0 / 4pi) * (I dl) / (R^2 + x^2)$.
By symmetry, the components of dB perpendicular to the axis cancel out, leaving only the axial components dB cos(phi), where phi is the angle between r and the axis.
Substitute cos(phi) = R / sqrt(R^2 + x^2) into the integral.
Integrate dl around the loop: $B = integral (\mu_0 I dl R) / (4pi (R^2 + x^2)^(3/2))$, noting that integral dl = 2pi R.
a = perpendicular distance, wire extends to infinity in one direction Tesla (T)
Semi-infinite straight conductor
Permeability of Free Space Constant
$$\mu_0 = 4\pi \times 10^{-7} \text{ T m A}^{-1}$$
m$u_0 = magnetic permeability of vacuum$ T m A^-1 or H m^-1
Free space / vacuum standard value
Graphs and experiments
Magnetic Field Variation Along the Axis of a Circular Coil
Draw: A bell-shaped symmetrical curve of B versus axial distance x, showing a maximum at x = 0 and inflection points at $x = +/- R/2$.
Interpret: The magnetic field is maximum at the center of the loop and drops off asymptotically as distance x increases.
Exam use: Frequently tested for identifying inflection points and slope properties in PGT examinations.
Right-Hand Thumb Rule / Maxwell's Corkscrew Rule Verification
Draw: A straight wire with concentric circular magnetic field lines with directional arrows.
Interpret: Visualizes the transverse nature of the magnetic field produced by linear current elements.
Exam use: TGT conceptual questions regarding direction of magnetic fields.
Helmholtz Coils Configuration
Draw: Two identical coaxial circular coils separated by a distance equal to their radius R, showing a flat region of uniform magnetic field in between.
Interpret: Demonstrates superposition of magnetic fields to achieve a region of constant magnetic field gradient.
Exam use: Standard PGT conceptual question on uniformity of magnetic field.
Solved numericals
TGT core
Calculate the magnetic field at the center of a circular coil of radius 10 cm containing 100 turns, carrying a current of 2 A.
Identify given values: Radius R = 10 cm = 0.1 m, Number of turns N = 100, Current I = 2 A.
Use formula for circular coil center: $B = (\mu_0 N I) / (2R)$.
A circular coil of radius R = 0.05 m has 50 turns and carries a current of 1 A. Find the magnetic field at an axial point at a distance x = 0.12 m from the center of the coil.
Identify given values: R = 0.05 m, x = 0.12 m, N = 50, I = 1 A.
Assumes steady electric currents (dc) where charge distribution does not vary with time; fails for time-varying displacement currents unless modified by Maxwell's equations.
Applicable strictly in free space or homogeneous media where magnetic permeability mu can be treated as a constant scalar value.
Common misconceptions
Common error: The magnetic field due to a current element is parallel to the current element.
Correct: The magnetic field is perpendicular to both the current element dl and the position vector r due to the cross product dl x r-hat.
Misinterpreting dot products with cross product operations in vector analysis.
Common error: Biot-Savart law can only be applied to closed circuits.
Correct: The differential form applies to any individual current element, though isolated open current elements cannot exist physically in steady state.
Confusing the physical existence of an isolated element with the mathematical validity of integrating differential elements.
Quick revision
Biot-Savart law gives dB proportional to I dl sin(theta) / r^2 with constant mu_0 / 4pi.
The SI value of the permeability of free space $\mu_0$ is $4\pi \times 10^{-7} \text{ T m A}^{-1}$.
The direction of the magnetic field contribution $dB$ is perpendicular to the plane containing the current element $dl$ and the position vector $r$ via the right-hand rule.
The magnetic field at the center of a circular current loop of radius $R$ carrying current $I$ is given by $B = \frac{\mu_0 I}{2R}$.
The magnetic field at an axial distance $x$ from the center of a circular loop is $B = \frac{\mu_0 I R^2}{2(R^2 + x^2)^{3/2}}$.
For an infinitely long straight wire, the magnetic field at a perpendicular distance $a$ is $B = \frac{\mu_0 I}{2\pi a}$.
Coulomb's electrostatic field is conservative ($\nabla \times E = 0$), whereas the magnetostatic field is solenoidal ($\nabla \cdot B = 0$).
Glossary
Biot-Savart LawA physical law describing the magnetic field generated by an electric current element, expressed as $dB = \frac{\mu_0}{4\pi} \frac{I dl \times \hat{r}}{r^2}$.
Permeability of Free Space ($\mu_0$)A physical constant representing the magnetic permeability in a classical vacuum, equal to $4\pi \times 10^{-7} \text{ T m A}^{-1}$.
Current Element ($I dl$)A differential vector segment of a conductor carrying a steady electric current $I$.
Solenoidal FieldA vector field with a divergence of zero everywhere, characteristic of the magnetic field $B$ ($\nabla \cdot B = 0$).
Biot-Savart Law vs Coulomb's Law
Parameter
Biot-Savart Law
Coulomb's Law
Source Field
Vector current element ($I dl$)
Scalar electric charge ($q$)
Field Direction
Perpendicular to the plane of $dl$ and $r$
Along the line joining the charges
Distance Dependence
Inverse-square law ($1/r^2$)
Inverse-square law ($1/r^2$)
Field Nature
Solenoidal ($\nabla \cdot B = 0$)
Conservative ($\nabla \times E = 0$)
Practice Quiz
Choose an option to reveal the explanation.
1. What is the correct vector form of the Biot-Savart law for a current element $I dl$?
Explanation: The differential magnetic field $dB$ is given by the cross product of the current element $dl$ and the unit position vector $\hat{r}$, inversely proportional to $r^2$.
2. What is the exact SI unit and value of the permeability of free space $\mu_0$?
Explanation: The permeability of free space $\mu_0$ has an exact defined SI value of $4\pi \times 10^{-7} \text{ Tesla meter per Ampere (T m A}^{-1}$).
3. The CGS unit of magnetic field $B$ is:
Explanation: The SI unit of magnetic field is the Tesla (or $\text{Wb m}^{-2}$), whereas the CGS unit is the Gauss.
4. What is the magnitude of the magnetic field at the center of a circular loop of radius $R$ carrying current $I$?
Explanation: Integrating the Biot-Savart law over a full circular loop of radius $R$ yields $B = \frac{\mu_0 I}{2R}$ at the center.
5. The magnetic field at a perpendicular distance $a$ from an infinitely long straight wire carrying current $I$ is:
Explanation: For an infinitely long straight conductor, applying Ampere's law or integrating Biot-Savart law gives $B = \frac{\mu_0 I}{2\pi a}$.
6. What is the magnetic field at the end (semi-infinite limit) of a straight current-carrying wire at a perpendicular distance $a$?
Explanation: For a semi-infinite wire where one angle is $90^\circ$ and the other is $0^\circ$, the formula $B = \frac{\mu_0 I}{4\pi a}(\sin 90^\circ + \sin 0^\circ)$ simplifies to $\frac{\mu_0 I}{4\pi a}$.
7. The magnetostatic field is characterized by which vector calculus property?
Explanation: Magnetic field lines are continuous loops without endpoints, which implies that the divergence of $B$ is always zero (solenoidal field).
8. What is the magnetic field at an axial distance $x$ from the center of a circular loop of radius $R$ and $N$ turns?
Explanation: The axial magnetic field of a coil with $N$ turns is given by $B = \frac{\mu_0 N I R^2}{2(R^2 + x^2)^{3/2}}$.
9. Under what condition does the Biot-Savart law strictly apply?
Explanation: The Biot-Savart law is formulated for steady currents (magnetostatics) where charge density is invariant with time.
10. How does the magnetic field contribution $dB$ vary with distance $r$ from a current element?
Explanation: Like Coulomb's law, the Biot-Savart law features an inverse-square dependence on distance $r$ ($1/r^2$).
11. What is the angle $\theta$ between the current element $dl$ and position vector $r$ that yields the maximum magnetic field contribution $dB$?
Explanation: Since $dB$ depends on $\sin\theta$, the magnitude is maximized when $\theta = 90^\circ$ (sin $90^\circ = 1$).
12. Which of the following statements correctly contrasts electrostatics and magnetostatics?
Explanation: Electrostatic fields have zero curl (conservative, $\nabla \times E = 0$), whereas magnetostatic fields have zero divergence (solenoidal, $\nabla \cdot B = 0$).
13. What is the magnetic field at the center of a circular arc of radius $R$ subtending an angle $\theta$ (in radians) at the center, carrying current $I$?
Explanation: For a full circle ($\theta = 2\pi$), $B = \frac{\mu_0 I}{2R}$. For an arc subtending angle $\theta$, it scales proportionally to $\frac{\theta}{2\pi}$, giving $\frac{\mu_0 I \theta}{4\pi R}$.
14. Why can an isolated magnetic monopole not exist according to the Biot-Savart law and Maxwell's equations?
Explanation: Steady currents exist only in closed paths, ensuring that magnetic field lines are continuous loops ($\nabla \cdot B = 0$), precluding magnetic monopoles.
15. If the current through a circular loop is doubled and its radius is halved, what happens to the magnetic field at its center?
Explanation: Using $B = \frac{\mu_0 I}{2R}$, if $I \to 2I$ and $R \to R/2$, then $B' = \frac{\mu_0 (2I)}{2(R/2)} = 4 \left(\frac{\mu_0 I}{2R}\right)$, so it increases by 4.
PYQ Patterns
These are pattern-based practice questions. Authentic year labels are only used when verified from a source.
1. A current $I$ flows along an infinitely long straight wire. The magnetic field at a perpendicular distance $a$ from the wire is proportional to:
Explanation: From the formula $B = \frac{\mu_0 I}{2\pi a}$, the magnetic field is inversely proportional to the distance $a$.
2. The magnetic induction at the center of a circular coil of radius $R$ carrying current $I$ is $B_1$. The magnetic induction on the axis of the same coil at a distance $x = R$ from the center is $B_2$. What is the ratio $B_1 / B_2$?
Explanation: $B_1 = \frac{\mu_0 I}{2R}$. At $x = R$, $B_2 = \frac{\mu_0 I R^2}{2(R^2 + R^2)^{3/2}} = \frac{\mu_0 I}{2 \sqrt{8} R} = \frac{\mu_0 I}{4\sqrt{2} R}$. Therefore, $B_1 / B_2 = \frac{\mu_0 I / 2R}{\mu_0 I / 4\sqrt{2}R} = 2\sqrt{2}$.
3. Two long parallel wires separated by a distance $d$ carry equal currents $I$ in the same direction. What is the magnetic field exactly midway between the wires?
Explanation: By the right-hand rule, currents in the same direction produce magnetic fields in opposite directions at the midpoint between them, canceling each other out to give zero.
4. A circular coil of $N$ turns and radius $R$ carries a current $I$. The magnetic field at its center is proportional to:
Explanation: The center magnetic field formula is $B = \frac{\mu_0 N I}{2R}$, showing direct proportionality to $N$ and inverse proportionality to $R$.
5. What is the angle between the position vector $r$ and the magnetic field vector $dB$ produced by a current element $I dl$?
Explanation: Since $dB$ is proportional to $dl \times \hat{r}$, the cross product ensures that $dB$ is perpendicular to both $dl$ and $\hat{r}$ (position vector), forming an angle of $90^\circ$.
6. A straight wire of finite length carrying current $I$ subtends angles $\phi_1$ and $\phi_2$ at a point $P$ at perpendicular distance $a$. If the wire is infinitely long, what do $\phi_1$ and $\phi_2$ approach?
Explanation: For an infinitely long straight wire, the ends extend to infinity, making the subtended angles $\phi_1$ and $\phi_2$ both approach $90^\circ$.
Timed Topic Test
Attempt these questions under a ten-minute limit using a separate timer.
1. The magnetic field at the center of a circular coil of radius $R$ is $B_0$. At what distance along the axis of the coil is the magnetic field reduced to $B_0 / 8$?
Explanation: We have $B_{axis} = \frac{\mu_0 I R^2}{2(R^2 + x^2)^{3/2}} = \frac{B_0}{(1 + x^2/R^2)^{3/2}} = \frac{B_0}{8}$. Thus, $(1 + x^2/R^2)^{3/2} = 8 \implies 1 + x^2/R^2 = 4 \implies x^2/R^2 = 3 \implies x = \sqrt{3}R$.
2. A current $I$ flows in a conductor shaped as a regular polygon of $n$ sides inscribed in a circle of radius $R$. What is the magnetic field at the center as $n \to \infty$?
Explanation: As $n$ approaches infinity, a regular polygon inscribed in a circle approaches a circle of radius $R$, whose center magnetic field is $\frac{\mu_0 I}{2R}$.
3. Two concentric coplanar circular loops of radii $R_1$ and $R_2$ ($R_1 < R_2$) carry currents $I_1$ and $I_2$ in opposite directions. For the net magnetic field at the center to be zero, what must be the ratio $I_1 / I_2$?
4. What is the line integral of the magnetic field $\oint B \cdot dl$ around a closed path enclosing a total steady current $I_{enc}$?
Explanation: Ampere's Circuital Law states that $\oint B \cdot dl = \mu_0 I_{enc}$ for steady currents.
5. A current-carrying circular loop is placed in a uniform magnetic field. Which orientation experiences maximum torque?
Explanation: Torque is given by $\tau = M \times B$, which has maximum magnitude when the angle between $M$ and $B$ is $90^\circ$ (perpendicular).
6. What is the magnetic field at the center of a square loop of side length $L$ carrying current $I$?
Explanation: Each side of length $L$ contributes $B_1 = \frac{\mu_0 I}{4\pi (L/2)} (\sin 45^\circ + \sin 45^\circ) = \frac{\mu_0 I}{2\pi L}(\sqrt{2})$. For 4 sides, $B = 4 \times \frac{\sqrt{2} \mu_0 I}{2\pi L} = \frac{2\sqrt{2} \mu_0 I}{\pi L}$.
7. Which of the following field configurations is physically impossible according to magnetostatics?
Explanation: Magnetic field lines must always form continuous closed loops (solenoidal field, $\nabla \cdot B = 0$), so they cannot start or terminate at isolated points in space.
8. In the Biot-Savart law expression, if the direction of the current element $dl$ is reversed while keeping the position vector $r$ same, what happens to $dB$?
Explanation: Since $dB \propto dl \times \hat{r}$, reversing $dl$ replaces $dl$ with $-dl$, which reverses the direction of the cross product vector $dB$.
9. A long straight wire carries a current of $10 \text{ A}$. What is the magnitude of the magnetic field at a distance of $5 \text{ cm}$ from the wire?
10. Why does the Biot-Savart law use the cross product $dl \times \hat{r}$ rather than a dot product?
Explanation: The cross product geometrically guarantees that the resulting magnetic field $dB$ is perpendicular to the plane formed by $dl$ and $\hat{r}$.