AP E&M Chapter 14-16 Part 1: Magnetic Fields
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Magnetic Field
The magnetic field (磁场, 磁感应强度) is a vector field, denoted by the symbol $\vec{B}$.
💡 $\vec{B}$ is a pseudovector.
A moving charge can produce a magnetic field, and it can also feel a force due to a magnetic field.
A current-carrying wire is an aggregate of moving charges. Therefore, it can produce and respond to a magnetic field as well.
| Produce $\vec{B}$ | Respond to $\vec{B}$ | |
|---|---|---|
| Moving Charge | $\vec{B} = \frac {\mu_0} {4\pi} \frac {\textcolor{goldenrod}{q \vec{v}} \times \hat{r}} {r^2}$ | $\vec{F} = \textcolor{goldenrod}{q \vec{v}} \times \vec{B}$ |
| Differential Length of Wire | $d\vec{B} = \frac {\mu_0} {4\pi} \frac {\textcolor{limegreen}{I d\vec{l}} \times \hat{r}} {r^2}$ | $d\vec{F} = \textcolor{limegreen}{I d\vec{l}} \times \vec{B}$ |
I’ll elaborate on these 4 formulas.
Magnetic Field due to a Point Charge
\[\boxed{ \vec{B} = \frac {\mu_0} {4\pi} \frac {\textcolor{goldenrod}{q \vec{v}} \times \hat{r}} {r^2} }\]$\vec{r}$ is defined as a position vector pointing from the location of the charge to the location where the magnetic field is calculated. $\hat{r}$ is the unit vector at this direction.
This is an experimental result.
$\mu_0$, the permeability of free space (vacuum permeability, 真空磁导率), is a constant just like $\varepsilon_0$.
\[\mu_0 \approx 4\pi \times 10^{-7} \ \ce{T\cdot m/A}\]where $\ce{T}$, tesla, is the unit for magnetic field.
Magnetic Force on a Point Charge
The magnetic force on a point charge is given by
\[\boxed{ \vec{F} = \textcolor{goldenrod}{q \vec{v}} \times \vec{B} }\]From this formula, we can say $1 \ce{T} = 1 \ce{N\cdot s/(C\cdot m)} = 1 \ce{N/(A\cdot m)}$.
The direction of magnetic force is always perpendicular to that of the velocity, so it never does work on the point charge.
The Lorentz force (洛伦兹力) is the sum of the force due to the electric field and the force due to the magnetic field:
\[\colorbox{#424242}{$ \color{white}\boxed{ \vec{F} = q (\vec{E} + \vec{v} \times \vec{B}) } $}\]Magnetic Field due to a Differential Length of Wire
Because of the principle of superposition, we only need to calculate $\sum q\vec{v}$ in the wire and then plug it into the yellow part of $\vec{B} = \frac {\mu_0} {4\pi} \frac {\textcolor{goldenrod}{q \vec{v}} \times \hat{r}} {r^2}$.
\[\begin{aligned} & \sum\limits_{i=1}^{dN} e\vec{v} \\ =& e \vec{v}_D dN \\ =& n e \vec{v}_D A dl \\ =& I d\vec{l} & I = n e v_D A \\ \end{aligned}\] \[\boxed{ d\vec{B} = \frac {\mu_0} {4\pi} \frac {\textcolor{limegreen}{I d\vec{l}} \times \hat{r}} {r^2} }\]This formula is called the Biot-Savart Law. It’s called a law because this formula was derived (by experiment and guess) earlier than $\vec{B} = \frac {\mu_0} {4\pi} \frac {\textcolor{goldenrod}{q \vec{v}} \times \hat{r}} {r^2}$.
Magnetic Force on a Differential Length of Wire
Through the same process, we can derive
\[\boxed{ d\vec{F} = \textcolor{limegreen}{I d\vec{l}} \times \vec{B} }\]Gauss’s Law for Magnetism
\[\colorbox{#424242}{$ \color{white}\boxed{ \displaystyle \oint \vec{B} \cdot d\vec{A} = 0 } $}\]This formula is similar to Gauss’s Law for Electricity. However, because there are no magnetic monopoles (磁单极子), $\rho_{\text{mag}} = 0$. Therefore, this law is also called Absence of magnetic monopoles (无磁单极子定律).
Example 14.1 Infinitely Long Wire
\[\begin{aligned} dB_z &= \frac {\mu_0} {4\pi} \frac {I dl} {(r^2 + y^2)} \sin \theta \\ &= \frac {\mu_0} {4\pi} \frac {I dy} {(r^2 + y^2)} \frac r {\sqrt{r^2 + y^2}} \\ &= \frac {\mu_0} {4\pi} \frac {Ir} {(r^2 + y^2)^{\frac 3 2}} dy \\ \end{aligned}\] \[\begin{aligned} B &= \int dBa \\ &= \frac {\mu_0 I r} {4\pi} \int_{-\infty}^{+\infty} \frac 1 {(r^2 + y^2)^{\frac 3 2}} dy \\ &= \frac {\mu_0 I r} {4\pi} \int_{-\frac \pi 2}^{\frac \pi 2} \frac 1 {(r^2 + r^2 \tan^2 \theta)^{\frac 3 2}} (r \sec^2 \theta d\theta) & y = r \tan \theta \\ &= \frac {\mu_0 I} {4\pi r} \int_{-\frac \pi 2}^{\frac \pi 2} \cos\theta d\theta \\ &= \frac {\mu_0 I} {2\pi r} \\ \end{aligned}\] \[\boxed{\vec{B} = \frac {\mu_0 I} {2\pi r} \hat{k}}\]What is the magnetic field at the point $(−r, 0)$ caused by an infinitely long wire carrying a current $I$ along the $y$-axis in the $+\hat{j}$-direction?
Example 14.2 The Force Between Parallel Wires
\[\begin{aligned} \frac {F_{\text{on 1}}} l &= I_1 B \\ &= \frac {\mu_0 I_1 I_2} {2\pi r} \end{aligned}\]Given two infinitely long parallel wires lying on the $xy$-plane, what is the force per length exerted on wire 1 by wire 2?
Ampere’s law
Ampere’s law is a restatement of the Biot-Savart law.
\[\boxed{ \oint \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{enclosed}} }\]It’s a line integral. The positive direction of $I_{\text{enc}}$ is determined by the right-hand rule.
Ampere’s Law is not complete - it is only valid when an electric field is static. We will soon generalize it to Ampere-Maxwell law to deal with situations where this is not the case.
I’m not gonna prove it because it’s as complicated as Gauss’s Law for Electricity 😒.
Example 14.4
Solve Example 14.1 with Ampere’s law.
Use a circular Amperian path that lies in a plane perpendicular to the wire.
\[\begin{aligned} \oint \vec{B} \cdot d\vec{l} &= \mu_0 I_{\text{enc}} \\ (2\pi r) B &= \mu_0 I \\ B &= \frac {\mu_0 I} {2\pi r} \\ \end{aligned}\]