Physics Problems & Solutions: Electromagnetism

A segment of wire is bent into an arc of radius R and subtended angle θ, as shown in the figure. Point P is at the center of the circular segment. The wire carries current I. What is the magnitude of the magnetic field at P?

A. 0
B. ^μ₀I^θ/₍₂_π₎_²_R

C. ^μ₀I^θ/₄_πR
D. ^μ₀I^θ/₄_πR_²
E. ^μ₀I/₂_θ_R_²

(GR0177 #88)

Solution:

Biot Savart’s Law:

$\begin{aligned} d \vec{B}&=\frac{\mu_0}{4\pi} \frac{Id \vec{l}\times \hat{r}}{r^2} \\\\\hat{r}&=\vec{r}/r \\d \vec{B}&=\frac{\mu_0}{4\pi} \frac{Id \vec{l}\times \vec{r}}{r^3} \end{aligned}$

$\begin{aligned} &Id \vec{l}\perp \vec{R} \rightarrow Id \vec{l}\times\vec{R}=IR dl \\ &d \vec{B}=\frac{\mu_0}{4\pi} \frac{IR dl}{R^3} \\&dl=Rd\theta\rightarrow d \vec{B}=\frac{\mu_0}{4\pi} \frac{IR^2d\theta}{R^3} \\&B=\frac{\mu_0 I\theta}{4\pi R} \end{aligned}$

Answer: C

Physics Problems & Solutions

Electromagnetism - Biot Savart's Law

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