32-1 Gauss’ Law for Magnetic Fields

32-1 Gauss’ Law for Magnetic Fields#

Prompts

  • Explain why the net magnetic flux through any closed Gaussian surface is zero, and how this differs from Gauss’s law for electric fields.

  • What is the simplest magnetic structure that can be enclosed? Why can’t we have isolated magnetic poles (monopoles)?

  • Walk me through calculating the magnetic flux \(\Phi_B\) through a surface when the magnetic field \(\vec{B}\) is uniform and perpendicular to the surface. Then show how to set up the integral when \(\vec{B}\) varies.

  • I’m given a cylindrical surface in a magnetic field. How do I use Gauss’s law for magnetism to find the net outward flux? Can you work through an example?

Lecture Notes#

Overview#

  • Electric vs magnetic Gauss’ law — same structure, different physics

    • Electric: \(\Phi_E = \oint \vec{E} \cdot d\vec{A} = q_{\text{enc}}/\varepsilon_0\) (flux \(\propto\) enclosed charge)

    • Magnetic: \(\Phi_B = \oint \vec{B} \cdot d\vec{A} = 0\) (flux always zero)

  • Physical meaning: No net magnetic flux through any closed surface \(\Rightarrow\) no magnetic monopoles

Why “no monopoles”?

Electric charges can be isolated (\(+\) or \(−\)). Magnetic poles cannot: every magnet has both N and S. Breaking a magnet only gives smaller dipoles, never a single pole. Gauss’ law for \(\vec{B}\) encodes this.

Magnetic dipoles: the simplest magnetic structure#

  • Bar magnet = magnetic dipole

    • North pole: field lines diverge (source)

    • South pole: field lines converge (sink)

  • Every magnet — from bar magnets to atoms — is a dipole (or superposition of dipoles)

  • Break a magnet → each piece is still a dipole (N and S on each fragment)

Conceptual barrier: “Where are the poles?”

A Gaussian surface can cut through a magnet and seem to enclose only one pole. But field lines that enter the surface must come from somewhere — there is always an associated opposite pole. The net flux is still zero.

Gauss’ law for magnetic fields#

(206)#\[ \Phi_B = \oint_S \vec{B} \cdot d\vec{A} = 0 \]

  • Statement: Net magnetic flux through any closed Gaussian surface \(S\) is zero

  • Implication: No net “magnetic charge” anywhere in the space (otherwise, enclosing it by a closed surface would cause an violation)

  • Holds for: Any closed surface, any magnetic source (dipole, current loop, etc.)

Contrast with electric Gauss’ law#

Electric

Magnetic

Flux \(\propto\) enclosed charge

Flux = 0 always

Monopoles exist (electric charges)

Monopoles do not exist

\(\Phi_E \neq 0\) if \(q_{\text{enc}} \neq 0\)

\(\Phi_B = 0\) always

Physical intuition

Magnetic field lines form closed loops (no start or end). Electric field lines start on \(+\) and end on \(−\). That is why \(\Phi_B = 0\) for any closed surface: every line that enters also leaves.

Summary#

  • Simplest magnetic structure: dipole (N and S together)

  • Magnetic monopoles: do not exist

  • Gauss’ law for \(\vec{B}\): \(\Phi_B = 0\) for any closed surface

  • This law is the mathematical statement that there are no magnetic monopoles

Discussions#

Surface and line integrals#

We distinguish open vs closed manifolds (surfaces or curves): open ones have boundaries; closed ones do not. We use \(\int\) for open-manifold integrals and \(\oint\) for closed-manifold integrals.

Surface integral (flux): \(\Phi = \int_\Sigma \vec{F} \cdot d\vec{A}\)

  • Meaning: How much of the field “flows through” the surface. We sum \(\vec{F} \cdot \hat{n}\) over each small area \(dA\); \(\hat{n}\) is the outward normal vector.

  • Open surface \(\Sigma\): General case; the integral is called flux.

  • Closed surface \(S\): Denoted \(\oint_S \vec{F} \cdot d\vec{A}\) and called net flux (what enters minus what leaves).

Line integral: \(\int_\Gamma \vec{F} \cdot d\vec{s}\)

  • Meaning: How much the field contributes along the curve. We sum \(\vec{F} \cdot d\vec{s}\) along the path; \(d\vec{s}\) is tangent to the curve.

  • Open curve \(\Gamma\) (path from A to B): Line integral along the path. (Physical example: work done by a force along the path.)

  • Closed loop \(C\): Denoted \(\oint_C \vec{F} \cdot d\vec{s}\) and called circulation—how much the field “circulates around” an oriented loop.

Aspect

Surface integral

Line integral

Open

Flux \(\int_\Sigma \vec{F} \cdot d\vec{A}\)

Line integral \(\int_\Gamma \vec{F} \cdot d\vec{s}\)

Closed

Net flux \(\oint_S \vec{F} \cdot d\vec{A}\)

Circulation \(\oint_C \vec{F} \cdot d\vec{s}\)

Element

\(d\vec{A} = \hat{n}\,dA\) (\(\hat{n}\) = normal direction)

\(d\vec{s} = \hat{\ell}\,ds\) (\(\hat{\ell}\) = tangent direction)

Domain

Surface (2D)

Curve (1D)

In Maxwell

Gauss’s laws (closed \(S\))

Faraday, Ampère–Maxwell (closed \(C\))

Example 1: Flux — field at angle \(\theta\) from normal

Field: \(\vec{F} = F_0(\sin\theta\,\hat{x} + \cos\theta\,\hat{z})\) (constant magnitude \(F_0\), lies in \(xz\)-plane; \(\theta\) = angle from \(\hat{z}\)). Surface: rectangle in \(xy\)-plane, \(0 \le x \le a\), \(0 \le y \le b\). Outward normal \(\hat{n} = \hat{z}\).

Component form: \(d\vec{A} = \hat{z}\,dx\,dy\). Only the \(z\)-component of \(\vec{F}\) contributes:

\[ \vec{F} \cdot d\vec{A} = F_0\cos\theta \cdot dx\,dy. \]

Integral:

\[ \Phi = \int_0^a \int_0^b F_0\cos\theta\,dy\,dx = F_0\cos\theta \cdot ab. \]

Result: \(\Phi = F_0 ab\cos\theta\). When \(\theta = 0\), \(\Phi = F_0 ab\); when \(\theta = 90°\), \(\Phi = 0\). Only the normal component \(F_0\cos\theta\) contributes.

Example 2: Flux — spatially varying field (sinusoidal)

Field: \(\vec{F} = F_0 \sin(kx)\,\hat{y}\) (varies with \(x\); \(k\) = wave number). Surface: rectangle in \(xz\)-plane (normal \(\hat{n} = \hat{y}\)), \(0 \le x \le a\), \(0 \le z \le c\).

Component form: \(d\vec{A} = \hat{y}\,dx\,dz\). So \(\vec{F} \cdot d\vec{A} = F_0\sin(kx)\,dx\,dz\).

Integral (integrate \(z\) first, then \(x\)):

\[ \Phi = \int_0^a \int_0^c F_0\sin(kx)\,dz\,dx = F_0 c \int_0^a \sin(kx)\,dx = F_0 c \cdot \frac{1}{k}\bigl[-\cos(kx)\bigr]_0^a = \frac{F_0 c}{k}\bigl(1 - \cos(ka)\bigr). \]

Result: \(\Phi = \frac{F_0 c}{k}(1 - \cos(ka))\). The flux depends on how many “wavelengths” fit in \([0,a]\); if \(ka = 2\pi\), then \(\cos(ka)=1\) and \(\Phi=0\).

Example 3: Circulation — square in \(xy\)-plane, sinusoidal field

Field: \(\vec{F} = F_0 \sin(kx)\,\hat{y}\) (varies with \(x\)). Loop: square in \(xy\)-plane with corners \((0,0) \to (a,0) \to (a,b) \to (0,b) \to (0,0)\), traversed counterclockwise.

Side by side:

Side

Path

\(d\vec{s}\)

\(\vec{F}\)

\(\vec{F} \cdot d\vec{s}\)

Integral

Bottom

\(y=0\), \(x: 0\to a\)

\(\hat{x}\,dx\)

\(F_0\sin(kx)\hat{y}\)

\(0\)

\(0\)

Right

\(x=a\), \(y: 0\to b\)

\(\hat{y}\,dy\)

\(F_0\sin(ka)\hat{y}\)

\(F_0\sin(ka)\,dy\)

\(F_0 b\sin(ka)\)

Top

\(y=b\), \(x: a\to 0\)

\(-\hat{x}\,dx\)

\(F_0\sin(kx)\hat{y}\)

\(0\)

\(0\)

Left

\(x=0\), \(y: b\to 0\)

\(-\hat{y}\,dy\)

\(0\)

\(0\)

\(0\)

Result: \(\oint_C \vec{F} \cdot d\vec{s} = F_0 b\sin(ka)\). Only the right edge contributes, because there \(\vec{F}\) is parallel to \(d\vec{s}\) and nonzero. The circulation depends on \(a\) through \(\sin(ka)\).

Example 4: Circulation — square, uniform field at angle

Field: \(\vec{F} = F_0(\cos\alpha\,\hat{x} + \sin\alpha\,\hat{y})\) (uniform in \(xy\)-plane, angle \(\alpha\) from \(x\)-axis). Loop: same square as Example 3.

Side by side:

Side

\(d\vec{s}\)

\(\vec{F} \cdot d\vec{s}\)

Integral

Bottom

\(\hat{x}\,dx\)

\(F_0\cos\alpha\,dx\)

\(F_0 a\cos\alpha\)

Right

\(\hat{y}\,dy\)

\(F_0\sin\alpha\,dy\)

\(F_0 b\sin\alpha\)

Top

\(-\hat{x}\,dx\)

\(-F_0\cos\alpha\,dx\)

\(-F_0 a\cos\alpha\)

Left

\(-\hat{y}\,dy\)

\(-F_0\sin\alpha\,dy\)

\(-F_0 b\sin\alpha\)

Result: \(\oint_C \vec{F} \cdot d\vec{s} = 0\). For a uniform field, the contributions from opposite sides cancel. Contrast with Example 3, where the field varies along the path.

Example 5: Infinitesimal cube — divergence from net flux

Setup: Small cube with one corner at \((x,y,z)\) and sides \(dx\), \(dy\), \(dz\). Field \(\vec{F} = (F_x, F_y, F_z)\) varies with position.

Face-by-face flux (outward normal; to first order in \(dx, dy, dz\)):

Face

\(\hat{n}\)

\(\vec{F} \cdot \hat{n}\) (evaluated at face)

Area

Flux

\(x+dx\)

\(\hat{x}\)

\(F_x(x+dx, y, z)\)

\(dy\,dz\)

\(F_x(x+dx,y,z)\,dy\,dz\)

\(x\)

\(-\hat{x}\)

\(-F_x(x, y, z)\)

\(dy\,dz\)

\(-F_x(x,y,z)\,dy\,dz\)

\(y+dy\)

\(\hat{y}\)

\(F_y(x, y+dy, z)\)

\(dx\,dz\)

\(F_y(x,y+dy,z)\,dx\,dz\)

\(y\)

\(-\hat{y}\)

\(-F_y(x, y, z)\)

\(dx\,dz\)

\(-F_y(x,y,z)\,dx\,dz\)

\(z+dz\)

\(\hat{z}\)

\(F_z(x, y, z+dz)\)

\(dx\,dy\)

\(F_z(x,y,z+dz)\,dx\,dy\)

\(z\)

\(-\hat{z}\)

\(-F_z(x, y, z)\)

\(dx\,dy\)

\(-F_z(x,y,z)\,dx\,dy\)

Taylor expansion (e.g. \(F_x(x+dx,y,z) - F_x(x,y,z) \approx \frac{\partial F_x}{\partial x}dx\)):

\[ \oint_S \vec{F} \cdot d\vec{A} = \frac{\partial F_x}{\partial x}dx\,dy\,dz + \frac{\partial F_y}{\partial y}dx\,dy\,dz + \frac{\partial F_z}{\partial z}dx\,dy\,dz = \left(\frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}\right)dx\,dy\,dz. \]

Result: \(\oint_S \vec{F} \cdot d\vec{A} = \mathsf{div} \vec{F} \,dx\,dy\,dz\), where

\[ \mathsf{div} \vec{F}:=\nabla \cdot \vec{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}. \]

Example 6: Infinitesimal rectangle — curl from circulation

Setup: Small rectangle in \(xy\)-plane with corners \((x,y)\), \((x+dx,y)\), \((x+dx,y+dy)\), \((x,y+dy)\), traversed counterclockwise. Field \(\vec{F} = (F_x, F_y, F_z)\).

Side-by-side (only \(F_x\), \(F_y\) matter; \(d\vec{s}\) is in \(xy\)-plane):

Side

Path

\(d\vec{s}\)

\(\vec{F} \cdot d\vec{s}\)

Integral

Bottom

\(y\) fixed

\(\hat{x}\,dx\)

\(F_x(x,y)\,dx\)

\(F_x(x,y)\,dx\)

Right

\(x+dx\) fixed

\(\hat{y}\,dy\)

\(F_y(x+dx,y)\,dy\)

\(F_y(x+dx,y)\,dy\)

Top

\(y+dy\) fixed

\(-\hat{x}\,dx\)

\(-F_x(x,y+dy)\,dx\)

\(-F_x(x,y+dy)\,dx\)

Left

\(x\) fixed

\(-\hat{y}\,dy\)

\(-F_y(x,y)\,dy\)

\(-F_y(x,y)\,dy\)

Net circulation:

\[ \oint_C \vec{F} \cdot d\vec{s} = \bigl[F_y(x+dx,y) - F_y(x,y)\bigr]dy - \bigl[F_x(x,y+dy) - F_x(x,y)\bigr]dx. \]

Taylor expansion: \(F_y(x+dx,y) - F_y(x,y) \approx \frac{\partial F_y}{\partial x}dx\); \(F_x(x,y+dy) - F_x(x,y) \approx \frac{\partial F_x}{\partial y}dy\). So

\[ \oint_C \vec{F} \cdot d\vec{s} = \frac{\partial F_y}{\partial x}dx\,dy - \frac{\partial F_x}{\partial y}dx\,dy = \left(\frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y}\right)dx\,dy. \]

Result: \(\oint_C \vec{F} \cdot d\vec{s} = (\mathsf{curl} \vec{F})_z\,dx\,dy\), where the \(z\)-component of the curl is

\[ (\mathsf{curl} \vec{F})_z:=(\nabla \times \vec{F})_z = \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y}. \]