Magnetic Field of Currents and Loops: Physics 10220 Tutorial 7 Study Notes

Magnetic Fields Due to Currents

Magnetic Field of a Wire Bent into an Arc

When a current-carrying wire is shaped into an arc, the magnetic field at the center of curvature can be determined using the Biot-Savart Law. This law relates the magnetic field produced at a point by a small segment of current-carrying wire.

• Biot-Savart Law: The magnetic field $d\vec{B}$ at a point due to a small segment $d\vec{l}$ of current $I$ is given by: $d\vec{B} = \frac{\mu_0}{4\pi} \frac{I d\vec{l} \times \hat{r}}{r^2}$

• Arc of a Circle: For a wire bent into an arc of radius $R$ and subtending an angle $\theta$ at the center, the total magnetic field at the center is: $B = \frac{\mu_0 I \theta}{4\pi R}$ where $\mu_0$ is the permeability of free space, $I$ is the current, $R$ is the radius, and $\theta$ is in radians.

• Direction: The direction of the magnetic field is given by the right-hand rule: if the fingers of your right hand follow the direction of current, your thumb points in the direction of the magnetic field at the center.

• Example: For an arc with $R = 0.5$ m, $I = 2$ A, and $\theta = \frac{\pi}{2}$ radians: $B = \frac{(4\pi \times 10^{-7}) \times 2 \times (\pi/2)}{4\pi \times 0.5} = \frac{2 \times 10^{-7} \times \pi}{0.5}$

Additional info: The Biot-Savart Law is fundamental for calculating magnetic fields from arbitrary current distributions, especially when symmetry is limited.

Magnetic Flux Through a Loop Near a Straight Wire

Calculating Magnetic Flux

When a current-carrying straight wire is placed near a rectangular loop, the wire generates a magnetic field that passes through the loop. The total magnetic flux through the loop can be found by integrating the magnetic field over the area of the loop.

• Magnetic Field of a Straight Wire: The magnetic field at a distance $r$ from a long straight wire carrying current $I$ is: $B = \frac{\mu_0 I}{2\pi r}$

• Magnetic Flux ($\Phi_B$): The flux through the loop is: $\Phi_B = \int B \cdot dA$ For a rectangular loop parallel to the wire, integrate $B$ over the area: $\Phi_B = \int_{r_1}^{r_2} \frac{\mu_0 I}{2\pi r} l \, dr = \frac{\mu_0 I l}{2\pi} \ln\left(\frac{r_2}{r_1}\right)$ where $l$ is the length of the loop parallel to the wire, $r_1$ and $r_2$ are the distances from the wire to the near and far sides of the loop.

• Direction of Flux: The direction of the magnetic field through the loop is determined by the right-hand rule.

• Example: For $I = 5$ A, $l = 0.2$ m, $r_1 = 0.01$ m, $r_2 = 0.03$ m: $\Phi_B = \frac{4\pi \times 10^{-7} \times 5 \times 0.2}{2\pi} \ln\left(\frac{0.03}{0.01}\right)$

Additional info: This calculation is important in understanding mutual inductance and the basis for electromagnetic induction in circuits.

Summary Table: Key Equations

Situation	Equation	Variables
Arc of Circle (center)	$B = \frac{\mu_0 I \theta}{4\pi R}$	$I$ = current, $R$ = radius, $\theta$ = angle (radians)
Straight Wire (distance $r$)	$B = \frac{\mu_0 I}{2\pi r}$	$I$ = current, $r$ = distance from wire
Flux through Rectangular Loop	$\Phi_B = \frac{\mu_0 I l}{2\pi} \ln\left(\frac{r_2}{r_1}\right)$	$I$ = current, $l$ = length of loop, $r_1$, $r_2$ = distances from wire