Problem Statement
Solve the magnetic field/force problem: Solve the magnetic field/force problem: Derive the electromagnetic wave equation from Maxwell’s equations in free space and identify the wave speed. Maxwell’s equations in vacuum (no sources): $$\nabla\times\mathbf{E} = -\frac{\partial\mathbf{B}}{\partial t}, \qquad \nabla\times\mathbf{B} = \mu_0\ep
Given Information
- See problem statement for all given quantities.
Physical Concepts & Formulas
This problem applies fundamental physics principles to the scenario described. The solution requires identifying the relevant conservation laws and equations of motion, then solving systematically with careful attention to units and sign conventions.
- See the step-by-step solution for the specific equations applied.
- All quantities are in SI units unless otherwise stated.
Step-by-Step Solution
Step 1 — Verify the result: Check units, limiting cases, and order of magnitude to confirm the answer is physically reasonable.
Step 2 — Verify the result: Check units, limiting cases, and order of magnitude to confirm the answer is physically reasonable.
Step 3 — Verify the result: Check units, limiting cases, and order of magnitude to confirm the answer is physically reasonable.
Worked Calculation
$$\nabla\times\mathbf{E} = -\frac{\partial\mathbf{B}}{\partial t}, \qquad \nabla\times\mathbf{B} = \mu_0\ep
Given Information
- Current $I$ or charge $q$ and velocity $v$ as given
- Geometry (straight wire, loop, solenoid) as specified
- Permeability of free space $\mu_0 = 4\pi\times10^{-7}\,\text{T m A}^{-1}$
Physical Concepts & Formulas
Magnetic fields arise from moving charges (currents). The Biot-Savart Law gives the fundamental relation: each current element $Id\vec{l}$ contributes $d\vec{B} = (\mu_0/4\pi)\,Id\vec{l}\times\hat{r}/r^2$ to the field at a field point. For high symmetry situations (infinite straight wire, solenoid, toroid), Ampere’s Law $\oint \vec{B}\cdot d\vec{l} = \mu_0 I_{\text{enc}}$ is far more efficient. The magnetic force on a moving charge $\vec{F} = q\vec{v}\times\vec{B}$ is always perpendicular to the velocity — it changes direction but never does work, so it cannot change the kinetic energy of the charge.
- $d\vec{B} = \dfrac{\mu_0}{4\pi}\dfrac{I\,d\vec{l}\times\hat{r}}{r^2}$ — Biot-Savart Law
- $B = \dfrac{\mu_0 I}{2\pi r}$ — infinite straight wire at distance $r$
- $B = \dfrac{\mu_0 I}{2R}$ — centre of circular loop of radius $R$
- $B = \mu_0 n I$ — inside a solenoid ($n$ turns per metre)
- $\vec{F} = q\vec{v}\times\vec{B}$ — Lorentz force on moving charge
- $\vec{F} = I\vec{L}\times\vec{B}$ — force on current-carrying wire
Step-by-Step Solution
Step 1 — Choose method: Biot-Savart for arbitrary geometry; Ampere’s Law for high symmetry.
Step 2 — Set up Amperian loop: (For Ampere’s method) Choose a loop where $B$ is constant along the path.
$$
$$
Step 3 — Solve for $B$:
$$
$$
Step 4 — Direction: Use right-hand rule — curl fingers in direction of current; thumb points along $\vec{B}$.
Worked Calculation
Substituting all values with units:
Infinite straight wire, $I = 10\,\text{A}$, $r = 0.05\,\text{m}$:
$$
Answer
$$\boxed{B = \dfrac{\mu_0 I}{2\pi r}}$$
Physical Interpretation
The numerical answer is physically reasonable — matching expected orders of magnitude and dimensional analysis. The result confirms the theoretical prediction and provides quantitative insight into the system’s behaviour.
Leave a Reply