Problem Statement
Find the magnetic field at the centre of a circular loop of radius $R = 9\,\text{cm}$ carrying current $I = 4\,\text{A}$.
Given Information
- $R = 0.09\,\text{m}$
- $I = 4\,\text{A}$
- $\mu_0 = 4\pi\times10^{{-7}}\,\text{{T m A}}^{{-1}}$
Physical Concepts & Formulas
By Biot–Savart law, every current element of the loop contributes a field at the centre in the same direction. Integrating: $B = \mu_0 I/(2R)$.
- $B = \mu_0 I/(2R)$ — centre of circular loop
Step-by-Step Solution
Step 1 — Formula: $B = \mu_0 I/(2R)$
Step 2 — Substitute: $$B = \frac{4\pi\times10^{-7}\times4}{2\times0.09}$$
Step 3 — Compute: $$B = 2.79e-05\,\text{T}$$
Worked Calculation
$$B = \frac{4\pi\times10^{-7}\times4}{2\times0.09} = 2.79e-05\,\text{T}$$
Answer
$$\boxed{B = 2.79e-05\,\text{T}}$$
The field 2.79e-05 T at the centre of this 9 cm radius loop. Helmholtz coil pairs use two such loops to create uniform fields for physics experiments.
Physical Interpretation
The field 2.79e-05 T at the centre of this 9 cm radius loop. Helmholtz coil pairs use two such loops to create uniform fields for physics experiments.
The magnitude of this result is physically reasonable and consistent with the expected order of magnitude for this class of problem. Comparing with standard values from physical tables confirms we are in the correct range.
This problem illustrates a fundamental principle that appears throughout physics: small changes in one parameter can lead to measurable, predictable changes in the observable quantity. Understanding this relationship is key to experimental design.
Note that the result depends on the square (or square root) of the key variable — this nonlinear dependence is characteristic of many physics phenomena and means that doubling the parameter does not simply double the result. Students should always check dimensional consistency and order-of-magnitude before accepting any numerical answer.
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