E&M Griffiths Chapter 5 Magnetostatics

What is the fundamental law that describes the force on a charged particle in electromagnetic fields?

The Lorentz Force Law: {\bf F} = q({\bf E} + {\bf v} \times {\bf B})

True or False: Magnetic fields can do work on charged particles.

False
— they only change the direction of motion, not the speed, since the force is always perpendicular to velocity.

What is the SI unit of the magnetic field (B-field)?

Tesla (T), equivalent to N·s/(C·m) or Wb/m².

How is a magnetic field defined in terms of the Lorentz force?

{\bf B} is defined by {\bf F} = q({{\bf v} \times {\bf B}}) for a charge moving in a region with no electric field.

What are the key differences between electric and magnetic fields?

Electric fields act on stationary charges and do work; magnetic fields act only on moving charges and do no work.

What produces magnetic fields in nature?

Moving charges (currents) or intrinsic magnetic dipole moments of particles (e.g., electrons).

How does the magnetic force on a moving charge depend on its direction?

{\bf F} = qvB\sin\theta; force is maximum when velocity is perpendicular to B, zero when parallel.

What is the direction of magnetic force on a positive charge moving through a magnetic field?

Given by the right-hand rule: fingers in direction of velocity, curl toward B, thumb gives direction of force.

What is a current element in the Biot–Savart law?

A vector quantity I{\bf dl} representing current I flowing through an infinitesimal length {\bf l}.

Write the Biot–Savart Law for the magnetic field of a steady current.

{\bf B}({\bf r}) = (\mu_0/4\pi) \int (I {\bf dl} \times \hat{r}) / r^2

In the Biot–Savart law, what does the cross product signify physically?

The magnetic field direction is perpendicular to both the current element and the displacement vector.

What is meant by a “steady current”?

A current that is constant in time; charge density and current distribution do not change with time.

Why is \nabla\cdot{\bf B} = 0 in magnetostatics?

Because magnetic monopoles have never been observed — magnetic field lines are continuous loops.

What does \nabla\times{\bf B} = \mu_0{\bf J} represent physically?

Ampère’s Law in differential form; it relates circulating magnetic fields to the current density that produces them.

How is the integral form of Ampère’s Law written?

\oint {\bf B}\cdot{\bf dl} = \mu0 I{\text{enc}}, where I_{\text{enc}} is the total current passing through the loop.

What kind of symmetry makes Ampère’s Law most useful?

Highly symmetric situations — cylindrical, planar, or solenoidal symmetry.

For a long straight wire carrying current I, what is the magnetic field at distance r?

B = \mu_0 I / (2\pi r)

What is the direction of the magnetic field around a straight current-carrying wire?

Given by the right-hand rule — curl fingers around wire in direction of current, thumb points along I.

Compare electrostatics and magnetostatics in terms of field divergence.

Electrostatics: \nabla\cdot{\bf E} = \rho/\varepsilon_0; Magnetostatics: \nabla\cdot{\bf B} = 0.

Compare the curl of electric and magnetic fields.

\nabla\times{\bf E} = 0 (static case) (\rightarrow) conservative field; \nabla\times{\bf B} = \mu_0{\bf J} (\rightarrow) field generated by steady current.

What is the magnetic vector potential {\bf A} defined as?

A vector field where {\bf B} = \nabla\times{\bf A}.

Why is the vector potential not unique?

Because adding the gradient of any scalar function ({\bf A}' = {\bf A} + \nabla\lambda) gives the same B (gauge freedom).

What is a common gauge condition for simplifying magnetostatics problems?

The Coulomb gauge: \nabla\cdot{\bf A} = 0.

How is the vector potential related to the Biot–Savart law?

{\bf A}({\bf r}) = (\mu_0/4\pi) \int ({{\bf J}({\bf r}')}/|{{\bf r}-{\bf r}'}|) d\tau'.

What boundary conditions apply to magnetic fields at a surface current {\bf K}?

The tangential component of {\bf B} changes by \mu_0{\bf K}\times\hat{n} across the boundary; normal component is continuous.

True or False: The normal component of {\bf B} is always discontinuous across a boundary.

False — it is continuous since \nabla\cdot{\bf B} = 0 implies no “magnetic charge.”

What is the physical meaning of the vector potential {\bf A}?

It is a field whose curl gives the magnetic field, often visualized as describing the “flow” that generates magnetic lines.

What is the multipole expansion of the vector potential used for?

Approximating the magnetic field far from a localized current distribution using dipole, quadrupole, etc. terms.

What is the magnetic dipole moment of a current loop?

{\bf m} = I{\bf a} = I (area vector), direction by right-hand rule.

What is the magnetic field of a dipole at a point far away?

{\bf B} = (\mu_0/4\pi r^3)[3({{\bf m}\cdot\hat{r}})\hat{r} - {\bf m}]

What is the analogy between magnetic dipoles and electric dipoles?

Both have fields falling off as 1/r^3 and produce similar angular dependencies.

True or False: The magnetic flux through a closed surface is zero.

True — because magnetic field lines form closed loops (\nabla\cdot{\bf B} = 0).

Why does steady current imply time-independent magnetic fields?

Because charge conservation (\partial\rho/\partial t = 0) and continuity equation ensure no buildup or depletion of charge.

What is the relationship between {\bf B} and {\bf H} in free space?

{\bf B} = \mu_0{\bf H}

What are the key assumptions of magnetostatics?

Currents are steady (time-independent), charge density is constant, and \partial{\bf E}/\partial t = 0.

What vector calculus identity connects \nabla\times{\bf B} and \nabla\cdot{\bf B}?

\nabla\cdot(\nabla\times{\bf B}) = 0 (\rightarrow) consistent with \nabla\cdot{\bf B} = 0.

Why can magnetic field lines never begin or end?

Because magnetic monopoles do not exist; field lines always form continuous loops.

What does the integral \oint{\bf B}\cdot{\bf dl} physically represent?

The circulation of the magnetic field around a closed loop — proportional to current enclosed.

What does it mean physically that \nabla\times{\bf B} = \mu_0{\bf J}?

Currents generate curling magnetic fields — the more current, the stronger the circulation.

What is the superposition principle for magnetic fields?

The net magnetic field is the vector sum of fields produced by all individual currents.

How do magnetic and electric forces differ in how they affect energy?

Electric forces can change kinetic energy; magnetic forces cannot, since they are perpendicular to motion.

In what way are {\bf A} and {\bf B} similar to \phi and {\bf E} in electrostatics?

{\bf A} (\rightarrow) {\bf B} via curl; \phi (\rightarrow) {\bf E} via gradient. Both potentials generate their respective fields.

What is the typical falloff of magnetic field strength from a long wire with distance?

B \propto 1/r

What condition must the current density {\bf J} satisfy in steady state?

\nabla\cdot{{\bf J}} = 0, expressing charge conservation.

How does the magnetic field behave inside a solenoid with tightly wound coils?

Nearly uniform and parallel to the solenoid’s axis; negligible outside (for an ideal solenoid).

What is magnetic flux and how is it calculated?

\Phi_B = \int{{\bf B}\cdot{\bf da}}, measuring total magnetic field through a surface.

True or False: Ampère’s law holds only for steady currents.

True — the full version includes a time-varying electric term (Maxwell’s correction).

What is the relationship between {\bf B} and current direction for a circular loop?

Field lines form closed circles through the loop center, direction given by right-hand rule.

Why does \nabla\times{\bf B} = \mu_0{\bf J} imply magnetic field lines are solenoidal?

Because the curl produces closed circulation — no sources or sinks.

In the vector potential formulation, what is the physical meaning of \nabla\cdot{\bf A} = 0?

It’s a mathematical constraint (Coulomb gauge) ensuring only physically relevant components remain.

What are the typical symmetry cases where the Biot–Savart law is most useful?

For localized current loops, straight wires, or arcs where integration is feasible.

What kind of problems are best solved by Ampère’s law versus Biot–Savart?

Ampère’s law: symmetric configurations (infinite wire, solenoid). Biot–Savart: localized or asymmetric currents.

What happens to magnetic field strength if the distance from a wire doubles?

It halves (B \propto 1/r).

How is the force between two parallel currents determined?

Using Ampère’s force law: per unit length, F/L = (\mu0 I1 I_2)/(2\pi r).

What does the sign of F/L between parallel currents indicate?

If currents are in the same direction (\rightarrow) attraction; opposite direction (\rightarrow) repulsion.

What is the magnetic dipole moment of N turns of current I in a loop of area A?

m = NIA.

How does the magnetic field of a dipole fall off with distance?

As 1/r^3 for far-field points.

What boundary condition holds for magnetic vector potential A at infinity?

{\bf A} \rightarrow 0 as r \rightarrow \infty for finite current distributions.