What is the magnetic field at 0 due to current in the infinite wire forming a loop as shown in the following figure?

Question

What is the magnetic field at 0 due to current in the infinite wire forming a loop as shown in the following figure? (A) (μ0 I/4 π d)( cos φ1+ cos φ2) (B) (μ0/4 π) × (2 l/d) (C) (μ0 I/4 π d)( sin φ1+ sin φ2) (D) (μ0/4 π) × (I/d)

Tardigrade Admin · Accepted Answer

Here, required angles θ1 and θ2 are (90°-φ1) an (90°-φ2) ∴ B =(μ0 I/4 π d)[ sin (90°-φ1)+ sin (90°-φ2)] =(μ0 I/4 π d)[ cos φ1+ cos φ2]

Q. What is the magnetic field at 0 due to current in the infinite wire forming a loop as shown in the following figure?

$\frac{μ _{0} I}{4 π d} (cos ϕ_{1} + cos ϕ_{2})$

$\frac{μ _{0}}{4 π} \times \frac{2 l}{d}$

$\frac{μ _{0} I}{4 π d} (sin ϕ_{1} + sin ϕ_{2})$

$\frac{μ _{0}}{4 π} \times \frac{I}{d}$

Here, required angles $θ_{1}$ and $θ_{2}$
are $(9 0^{\circ} - ϕ_{1})$ an
$(9 0^{\circ} - ϕ_{2})$
$∴ B = \frac{μ _{0} I}{4 π d} [sin (9 0^{\circ} - ϕ_{1}) + sin (9 0^{\circ} - ϕ_{2})]$
$= \frac{μ _{0} I}{4 π d} [cos ϕ_{1} + cos ϕ_{2}]$

Q. What is the magnetic field at 0 due to current in the infinite wire forming a loop as shown in the following figure?

4πdμ0​I​(cosϕ1​+cosϕ2​)

4πμ0​​×d2l​

4πdμ0​I​(sinϕ1​+sinϕ2​)

4πμ0​​×dI​

Here, required angles θ1​ and θ2​ are (90∘−ϕ1​) an (90∘−ϕ2​) ∴B=4πdμ0​I​[sin(90∘−ϕ1​)+sin(90∘−ϕ2​)] =4πdμ0​I​[cosϕ1​+cosϕ2​]

$\frac{μ _{0} I}{4 π d} (cos ϕ_{1} + cos ϕ_{2})$

$\frac{μ _{0}}{4 π} \times \frac{2 l}{d}$

$\frac{μ _{0} I}{4 π d} (sin ϕ_{1} + sin ϕ_{2})$

$\frac{μ _{0}}{4 π} \times \frac{I}{d}$

Here, required angles $θ_{1}$ and $θ_{2}$
are $(9 0^{\circ} - ϕ_{1})$ an
$(9 0^{\circ} - ϕ_{2})$
$∴ B = \frac{μ _{0} I}{4 π d} [sin (9 0^{\circ} - ϕ_{1}) + sin (9 0^{\circ} - ϕ_{2})]$
$= \frac{μ _{0} I}{4 π d} [cos ϕ_{1} + cos ϕ_{2}]$