circular loop center field

Question

Field at center of circular loop radius R carrying current I?

RankGuru · Accepted Answer

1. **Biot-Savart Law**: $dB = \frac{\mu_0}{4\pi} \frac{I \, dl \sin 	heta}{r^2}$.
2. **Integration for Circle**: At the center, every current element $dl$ is at distance $R$ and is perpendicular to the radius ($ 	heta=90^\circ$).
3. **Step-by-Step**:
   - $B = \int \frac{\mu_0}{4\pi} \frac{I \, dl}{R^2} = \frac{\mu_0 I}{4\pi R^2} \int dl$
   - Since $\int dl = 2\pi R$ (circumference):
   - $B = \frac{\mu_0 I}{4\pi R^2} 	imes (2\pi R) = \frac{\mu_0 I}{2R}$.
4. **Conclusion**: The magnetic field strength at the center of the loop is **$\frac{\mu_0 I}{2R}$**.

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circular loop center field

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