PHYS 208 Lecture 16

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Biot-Savart Law

Equation for finding a magnetic field at any point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{r}} away from a current-carrying wire.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{B} = \frac{\mu_0}{4 \pi} \int \frac{I \mathrm{d}\vec{\ell} \times \hat{r}}{\left|\vec{r}\right|^2} }

For an infinitely long, straight current carrying wire

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{B}(r) = \frac{\mu_0 I}{2 \pi r} }

Alternate Right-Hand Rule

Thumb along current, fingers point in direction of magnetic field.

For oncoming current, magnetic field is always counter-clockwise

Force Between Parallel Wires

Two infinitely long wires are separated a distance Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d} with the currents Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I_1} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I_2} of both flowing in the same direction.

The magnetic field at each point along each wire point in opposing direction.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \vec{F} &= I \vec{\ell} \times \vec{B} \\ \vec{B} &= \frac{mu_0 I}{2 \pi d} \\ \frac{\vec{F}}{\ell} &= \mu_0 \frac{I_1 I_2}{2 \pi d} \end{align}}

• When the currents are parallel, the wires attract.
• When the currents are anti-parallel (in opposite direction), the wires repel.

Magnetic Field at Center of Current Loop

A circular loop of radius Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} has a current Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I} flowing CCW. What is the magnetic field at the center of the circle?

(Dr. Webb refused to simplify his equation due to the symmetry, so I have taken the liberty to do it for him)

At every point around the circumference, the magnetic field points out from the loop towards us, so Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{B}} is in the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{k}} direction.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \vec{B} &= \frac{\mu_0}{4 \pi} \int \frac{I \mathrm{d}\ell}{R^2} \hat{k} \\ &= \frac{\mu_0 I}{4 \pi R^2} \int \mathrm{d}\ell \hat{k} \\ &= \frac{\mu_0 I}{4 \pi R^2} \left(2 \pi R \right) \hat{k} \\ &= \frac{\mu_0 I}{2 R} \hat{k} \end{align}}

Now that was much easier to understand, involved fewer steps, and still arrived at the same answer.

For any fraction of a circle, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{B}} scales accordingly:

• Semicircle (half of a circle): Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{2} \vec{B} = \frac{\mu_0 I}{4 R} \hat{k}}
• Quarter circle: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{4} \vec{B} = \frac{\mu_0 I}{8 R} \hat{k}}

Ampere's law

Finding magnetic fields using symmetry:

Suppose we integrate a magnetic field "flowing" through a closed wire loop:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \oint \vec{B} \cdot \mathrm{d}\vec{\ell} = \mu_0 I_{en}}