Magnetostatics¶

Possible Exam Questions¶

Syllabus Focus¶

• Biot-Savart's law
• Ampere's circuital law
• Curl

1. Introduction to Magnetostatics¶

Magnetostatics deals with magnetic fields produced by steady currents. A current is steady when its magnitude and distribution do not vary with time.

In electrostatics, stationary charges produce electric fields. In magnetostatics, moving charges or steady currents produce magnetic fields.

Magnetic Field Quantities¶

For a linear, isotropic medium:

where \(\mu_r\) is relative permeability.

Current and Current Density¶

Current is the rate of flow of charge:

Volume current density:

Unit: A/m\(^2\)

For a conductor with conductivity \(\sigma\) under electric field \(\vec E\):

This is the point form of Ohm's law.

Common current densities:

2. Magnetic Force and Lorentz Force¶

Force on a Moving Charge¶

A charge \(Q\) moving with velocity \(\vec u\) in a magnetic field \(\vec B\) experiences force:

Magnitude:

where \(\theta\) is the angle between \(\vec u\) and \(\vec B\).

Important points:

• Magnetic force is perpendicular to both velocity and magnetic field.
• Magnetic force does no work because it is perpendicular to motion.
• Maximum force occurs when \(\theta = 90^\circ\).
• Zero force occurs when the charge moves parallel to \(\vec B\).

Lorentz Force¶

If both electric and magnetic fields are present:

Force on a Current-Carrying Conductor¶

For a conductor carrying current \(I\) in magnetic field \(\vec B\):

For a straight conductor of length \(L\) in uniform magnetic field:

Magnitude:

Torque on a Current Loop¶

For a current loop with \(N\) turns, area \(A\), and current \(I\):

where magnetic dipole moment is:

Magnitude:

This is the operating principle of moving-coil instruments and DC motors.

3. Biot-Savart's Law¶

Statement¶

Biot-Savart's law gives the magnetic field intensity produced at a point by a differential current element.

For a line current element \(I\,d\vec l\):

where:

• \(I\) is current in amperes.
• \(d\vec l\) is the differential length element in the direction of current.
• \(R\) is the distance from the current element to the observation point.
• \(\hat a_R\) is the unit vector from the current element to the observation point.

Magnetic flux density form:

Integral Forms¶

For a line current:

For a surface current:

For a volume current:

Direction of Magnetic Field¶

The direction is found using the right-hand rule:

• Thumb points in the direction of current.
• Curled fingers show direction of magnetic field around the conductor.

4. Standard Results from Biot-Savart's Law¶

4.1 Infinite Straight Current-Carrying Conductor¶

For an infinitely long wire carrying current \(I\), the magnetic field at radial distance \(\rho\) is:

The field circles the wire and decreases as \(1/\rho\).

4.2 Finite Straight Conductor¶

For a finite straight conductor, if the observation point is at perpendicular distance \(\rho\) and the conductor subtends angles \(\alpha_1\) and \(\alpha_2\) at that point:

For symmetric angles \(\alpha_1 = -\alpha\), \(\alpha_2 = \alpha\):

As \(\alpha \to 90^\circ\), the conductor becomes infinitely long and:

4.3 Circular Current Loop on Axis¶

For a circular loop of radius \(a\) carrying current \(I\), magnetic field intensity on its axis at distance \(z\) from the center is:

At the center of the loop, \(z = 0\):

For \(N\) turns:

4.4 Solenoid¶

For a long solenoid with \(N\) turns, length \(l\), and current \(I\):

where \(n = N/l\) is turns per unit length.

The magnetic field inside a long solenoid is nearly uniform; outside it is very small.

4.5 Toroid¶

For a toroid with \(N\) turns carrying current \(I\), at radius \(\rho\) inside the core:

The field is mostly confined inside the toroidal core.

5. Ampere's Circuital Law¶

Statement¶

Ampere's circuital law states that the line integral of magnetic field intensity around any closed path is equal to the net current enclosed by that path.

In terms of current density:

Physical Meaning¶

Ampere's law relates the circulation of magnetic field around a closed path to current passing through the surface bounded by that path.

• Electric charges are sources/sinks of electric flux.
• Currents are sources of magnetic field circulation.
• Magnetic field lines form closed loops around currents.

When Ampere's Law is Useful¶

Ampere's law is most useful when symmetry allows \(\vec H\) to be constant along the chosen Amperian path.

Application 1 - Infinite Straight Wire¶

Choose a circular Amperian path of radius \(\rho\) around the wire.

By symmetry:

and \(H_\phi\) is constant on the circular path.

Using Ampere's law:

Application 2 - Long Solenoid¶

For a long solenoid with \(n\) turns per meter and current \(I\), choose a rectangular Amperian path with one side inside the solenoid and one side outside.

The outside field is approximately zero, and the inside field is nearly uniform.

Enclosed current:

Therefore:

Application 3 - Toroid¶

For a toroid with \(N\) turns and current \(I\), choose a circular Amperian path of radius \(\rho\) inside the core.

Enclosed current is \(NI\).

Application 4 - Infinite Current Sheet¶

For an infinite sheet carrying surface current density \(\vec K = K\hat a_x\) on the plane \(z=0\), the magnetic field is uniform on both sides and opposite in direction.

Using Ampere's law:

The direction is obtained from the right-hand rule.

6. Curl and Point Form of Ampere's Law¶

Stokes' Theorem¶

Stokes' theorem relates the line integral of a vector field around a closed path to the surface integral of the curl of that field over any surface bounded by the path.

Apply Stokes' theorem to Ampere's circuital law:

Using Stokes' theorem:

Since this is true for any surface:

This is Ampere's law in differential form for magnetostatics.

Curl: Meaning¶

Curl measures the local rotation or circulation of a vector field.

• If \(\nabla \times \vec H \ne 0\), the magnetic field has local circulation.
• In magnetostatics, local circulation of \(\vec H\) is caused by current density \(\vec J\).
• Where there is no current density, \(\nabla \times \vec H = 0\), even though \(\vec H\) may still exist due to currents elsewhere.

Curl in Cartesian Coordinates¶

For:

Relation Between Curl and Current Density¶

If \(\vec H\) is known, current density can be found from:

Example:

If \(\vec H = 5x\hat a_y\,\text{A/m}\):

So:

7. Magnetic Flux and Gauss' Law for Magnetism¶

Magnetic Flux¶

Magnetic flux through a surface is:

Unit: weber (Wb)

Gauss' Law for Magnetism¶

Magnetic flux through any closed surface is zero:

Differential form:

Physical meaning:

• There are no isolated magnetic charges or monopoles in classical electromagnetics.
• Magnetic field lines always form closed loops.
• As much magnetic flux enters a closed surface as leaves it.

8. Magnetic Vector Potential¶

Since:

magnetic flux density can be expressed as the curl of another vector field:

where \(\vec A\) is called the magnetic vector potential.

For a line current:

For volume current density:

Vector potential is especially useful for solving radiation, waveguide, and antenna problems.

9. Magnetic Energy and Inductance¶

Magnetic Energy Density¶

Energy stored per unit volume in a magnetic field is:

For a linear medium:

Total magnetic energy:

Inductance¶

Inductance is the flux linkage per unit current.

Unit: henry (H)

Energy stored in an inductor:

For a long solenoid:

where:

• \(N\) = number of turns
• \(A\) = cross-sectional area
• \(l\) = length of solenoid

10. Magnetic Boundary Conditions¶

At the boundary between two magnetic media, magnetic fields satisfy boundary conditions.

Normal Component of \(\vec B\)¶

From \(\nabla \cdot \vec B = 0\):

The normal component of magnetic flux density is continuous across the boundary.

Tangential Component of \(\vec H\)¶

From Ampere's law:

where \(\vec K_s\) is surface current density at the boundary.

If there is no surface current:

11. Solved Examples¶

Example 1 - Field Around a Long Straight Wire¶

Q. A long straight conductor carries \(10\,\text{A}\). Find \(H\) and \(B\) at a distance of \(5\,\text{cm}\) in free space.

Solution:

Example 2 - Magnetic Field at Center of Circular Coil¶

Q. A circular coil has 50 turns, radius \(10\,\text{cm}\), and current \(2\,\text{A}\). Find \(H\) at the center.

Solution:

Example 3 - Toroid Field¶

Q. A toroid has 400 turns and carries \(3\,\text{A}\). Find \(H\) at mean radius \(8\,\text{cm}\).

Solution:

Example 4 - Current Density from Curl of \(\vec H\)¶

Q. In a region, \(\vec H = (3y\hat a_x + 2x\hat a_y)\,\text{A/m}\). Find \(\vec J\).

Solution:

Only the \(z\)-component exists:

12. Quick Revision Table¶

Model Answer — Biot-Savart and Ampere Laws with Applications [5 marks]¶

Exam-ready answer

Assume steady currents in a linear medium. For an oriented source element \(I\,d\vec l'\) at \(\vec r'\), define \(\vec R=\vec r-\vec r'\), \(R=|\vec R|\), and \(\hat a_R\) from the source element to the field point. The Biot-Savart law is

\[ d\vec H=\frac{I\,d\vec l'\times\hat a_R}{4\pi R^2}, \qquad \boxed{\vec H(\vec r)=\frac{I}{4\pi}\int_C \frac{d\vec l'\times\hat a_R}{R^2}}, \]

where \([H]=\text{A/m}\) and \(\vec B=\mu\vec H\) in tesla. The cross product and right-hand rule set the field direction. For a distributed current, replace \(I\,d\vec l'\) by \(\vec J\,dv'\) or \(\vec K\,dS'\) as appropriate.

Ampere's circuital law states

\[ \boxed{\oint_C\vec H\cdot d\vec l=I_{\rm enc} =\int_S\vec J\cdot d\vec S}. \]

The positive normal to \(S\) is related to the positive traversal of \(C\) by the right-hand rule; a current through \(S\) along that normal is positive. Biot-Savart directly sums source contributions and works for arbitrary steady-current geometry, whereas Ampere's law becomes a practical field-solving method only when symmetry makes \(H\) constant or zero on sections of the path.

Biot-Savart application: at the centre of a circular loop of radius \(a\), every element has \(R=a\) and \(d\vec l'\perp\hat a_R\). All contributions point along the loop axis, so

\[ H_z=\frac{I}{4\pi a^2}\oint dl' =\frac{I}{4\pi a^2}(2\pi a) \quad\Rightarrow\quad \boxed{\vec H=\frac{I}{2a}\hat a_z}. \]

Ampere-law application: for a long solenoid of \(N\) turns and length \(\ell\), a rectangular path enclosing \(NI\) gives \(H\ell=NI\) because the outside field and transverse contributions are negligible:

\[ \boxed{\vec H\simeq nI\hat a_z}, \qquad n=N/\ell\;[\text{turns/m}]. \]

For example, \(N=500\), \(\ell=0.25\,\text{m}\) and \(I=0.10\,\text{A}\) give \(H=200\,\text{A/m}\). Finite solenoids have end fringing, and neither ideal result includes nonlinear saturation when \(\mu\) depends on \(H\). Applications include coils, electromagnets, current sensors and inductors.

Practice target: 9 minutes; state both orientation rules and derive one compact application of each law.

Model Answer — Ampere-Law Fields of a Wire, Solenoid, and Toroid [5 marks]¶

Exam-ready answer

For steady current, choose the contour direction first; the right-hand rule then defines the positive surface normal and enclosed current in

\[ \boxed{\oint_C\vec H\cdot d\vec l=I_{\rm enc}}, \qquad [H]=\text{A/m}. \]

Infinite straight conductor: let current \(I\) flow in \(+\hat a_z\). Cylindrical symmetry gives \(\vec H=H_\phi\hat a_\phi\), counter-clockwise when viewed from \(+z\). On a circular path of radius \(\rho\),

\[ H_\phi(2\pi\rho)=I \quad\Rightarrow\quad \boxed{\vec H=\frac{I}{2\pi\rho}\hat a_\phi}. \]

Long solenoid: for \(N\) turns over length \(\ell\), use a rectangular contour with one side inside along the axis. The outside field is approximately zero and the short sides are normal to \(\vec H\), hence

\[ H\ell=NI \quad\Rightarrow\quad \boxed{\vec H_{\rm inside}=nI\hat a_z}, \qquad n=N/\ell, \]

with direction set by curled fingers following winding current; \(H_{\rm outside}\simeq0\) away from the ends.

Toroid: for \(N\) turns carrying \(I\), choose a circular path of radius \(\rho\) within a core of inner radius \(a\) and outer radius \(b\). Symmetry gives azimuthal \(\vec H\):

\[ H_\phi(2\pi\rho)=NI \quad\Rightarrow\quad \boxed{\vec H=\frac{NI}{2\pi\rho}\hat a_\phi\quad(a<\rho<b)}. \]

For an ideal tightly wound toroid, \(H\simeq0\) in the central hole \(\rho<a\) and outside \(\rho>b\), because an appropriate spanning surface encloses zero net linked current. In each linear material, \(\vec B=\mu\vec H\) T.

Check: for \(I=2\,\text{A}\) at \(\rho=0.10\,\text{m}\), the wire field is \(H=2/(2\pi\times0.10)=3.18\,\text{A/m}\). Doubling distance halves it. The wire result has \(1/\rho\) variation, the ideal solenoid is nearly uniform, and the toroid varies as \(1/\rho\) but confines flux. Finite length, sparse winding, leakage, fringing and core saturation limit the ideal formulas.

Practice target: 9 minutes; draw each Amperian path, mark the positive current, and show which path sections contribute.

Model Answer — Curl and the Point Form of Ampere's Law [5 marks]¶

Model Answer — Scalar and Vector Magnetic Potentials [5 marks]¶

Model Answer — Magnetic Boundary Conditions [5 marks]¶

Exam-ready answer

Let an interface separate media 1 and 2, and define the unit normal \(\hat n\) to point from medium 1 into medium 2. Let a free surface-current sheet \(\vec K_s\) in A/m lie tangentially on the interface.

Normal component of \(\vec B\). Apply Gauss's law for magnetism to a thin pillbox of face area \(\Delta S\) and height tending to zero:

\[ \oint_S\vec B\cdot d\vec S=0. \]

Side flux vanishes with height. The upper face contributes \(B_{2n}\Delta S\) and the lower outward face contributes \(-B_{1n}\Delta S\), so

\[ (B_{2n}-B_{1n})\Delta S=0 \quad\Rightarrow\quad \boxed{\hat n\cdot(\vec B_2-\vec B_1)=0}, \]

or \(B_{2n}=B_{1n}\). There is no magnetic surface charge. For linear media this becomes

\[ \boxed{\mu_2H_{2n}=\mu_1H_{1n}}. \]

Tangential component of \(\vec H\). Apply Ampere's law to an infinitesimal rectangular loop straddling the boundary. The two short normal sides vanish. Choosing its traversal consistently with the right-hand rule, the enclosed sheet current is \(\vec K_s\) projected across the loop, giving the vector result

\[ \boxed{\hat n\times(\vec H_2-\vec H_1)=\vec K_s}. \]

Equivalently, for a tangent \(\hat t\) normal to \(\vec K_s\) in the surface, the signed jump is fixed by the chosen \(\hat n,\hat t\) orientation. If \(\vec K_s=0\),

\[ \boxed{\vec H_{2t}=\vec H_{1t}}. \]

Check: at a current-free boundary with \(\mu_2=4\mu_1\), continuity of normal \(B\) requires \(H_{2n}=H_{1n}/4\), while tangential \(H\) remains unchanged. Hence field lines refract according to \(\tan\theta_2/\tan\theta_1=\mu_2/\mu_1\) when \(\theta\) is measured from the normal. These conditions are used at air gaps, magnetic cores, shielding surfaces and waveguide walls. They assume an ideal zero-thickness interface; finite transition layers and nonlinear or hysteretic media require their local constitutive \(B(H)\) relation, but the integral boundary laws remain valid.

Practice target: 9 minutes; draw both infinitesimal constructions, mark the 1-to-2 normal, and retain the cross-product order.