The wire shown in Fig. 31-7 carries a current of 40 A. Find the field at point- P. Since P lies

step 1 In this given problem, a current carrying wire has been bent in the shape like this. The center

The wire shown in Fig. 31-7 carries a current of 40 A. Find...

The wire shown in Fig. $31-7$ carries a current of $40 \mathrm{~A}$. Find the field at point- $P$. Since $P$ lies on the lines of the straight wires, those wires contribute no field at $P$. A circular loop of radius $r$ gives a field of $B=\mu_{0} I / 2 r$ at its center point. Here we have only three-fourths of a loop, and so we can assume that $$\begin{aligned} B \text { at point- } P &=\left(\frac{3}{4}\right)\left(\frac{\mu_{0} I}{2 r}\right)=\frac{(3)\left(4 \pi \times 10^{-7} \mathrm{~T} \cdot \mathrm{m} / \mathrm{A}\right)(40 \mathrm{~A})}{(4)(2)(0.020 \mathrm{~m})} \\ &=9.4 \times 10^{-4} \mathrm{~T}=0.94 \mathrm{mT} \end{aligned}$$ The field is out of the page.

The wire shown in Fig. $31-7$ carries a current of $40 \mathrm{~A}$. Find the field at point- $P$.
Since $P$ lies on the lines of the straight wires, those wires contribute no field at $P$. A circular loop of radius $r$ gives a field of $B=\mu_{0} I / 2 r$ at its center point. Here we have only three-fourths of a loop, and so we can assume that
$$\begin{aligned}
B \text { at point- } P &=\left(\frac{3}{4}\right)\left(\frac{\mu_{0} I}{2 r}\right)=\frac{(3)\left(4 \pi \times 10^{-7} \mathrm{~T} \cdot \mathrm{m} / \mathrm{A}\right)(40 \mathrm{~A})}{(4)(2)(0.020 \mathrm{~m})} \\
&=9.4 \times 10^{-4} \mathrm{~T}=0.94 \mathrm{mT}
\end{aligned}$$
The field is out of the page.

Key Concepts

Magnetic Field of a Current Loop

A complete current loop generates a magnetic field at its center that is determined by the current passing through the loop, the radius of the loop, and the permeability of free space. The standard expression B = ??I/2r encapsulates this relationship and is fundamental in analyzing how currents produce magnetic fields in symmetric, closed-loop configurations.

Fractional Arc Contribution

When dealing with an arc or a segment of a current loop rather than a full circle, the magnetic field at the central point is scaled according to the fraction of the complete loop present. This idea stems from the linearity of the Biot–Savart law, allowing one to simply multiply the field of a complete loop by the fraction of the loop’s angle over 2? to determine the field produced by the arc.

Biot–Savart Law

The Biot–Savart law provides a detailed mathematical description of the magnetic field generated by a small element of current. It states that the contribution to the magnetic field from a current segment is proportional to the current element and inversely proportional to the square of the distance from the element. This law is essential for calculating magnetic fields produced by arbitrarily shaped current distributions.

Superposition Principle

Magnetic fields, being vector quantities, obey the superposition principle, which means that the net magnetic field created by several current-carrying pieces is the vector sum of the individual fields from each segment. This principle ensures that complex configurations can be analyzed by breaking them down into simpler components, calculating the magnetic field due to each, and then adding the results.

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