Theory — Magnetic Field of a Current Loop
Currents Make Magnetic Fields
A moving electric charge creates a magnetic field. When current flows around a loop of wire, the magnetic field from every part of the loop adds together and is strongest at the centre, pointing along the loop's axis. The direction is given by the right-hand rule: curl the fingers of the right hand in the direction of the current and the thumb points along the field.
Field at the Centre of a Loop
For a circular loop (or a tight coil of n turns) carrying current I, the magnetic field at the centre depends on the current, the number of turns, and the loop radius.
B = magnetic field at the centre (T)
μ₀ = 4π × 10⁻⁷ T·m/A (permeability of free space)
n = number of turns, I = current (A), R = loop radius (m)
How the Field Responds
Because B is proportional to n and to I, doubling either one doubles the field. Because B is inversely proportional to R, doubling the radius halves the field. The magnetic field is a vector — it has both a magnitude (measured in tesla) and a direction (along the axis of the loop).
Recovering the Radius
If you measure B for a known current and number of turns, you can rearrange the formula to solve for the loop radius. Since the loop itself does not change, the radius you calculate should come out the same for every trial — a good check that the relationship holds.
More turns
B ∝ n. Each extra turn adds its own field, so the centre field grows in proportion to n.
More current
B ∝ I. A bigger current means faster-moving charge and a stronger field.
Bigger loop
B ∝ 1/R. Spreading the current over a larger loop weakens the field at the centre.
| Quantity | Relationship | Notes |
|---|---|---|
| Centre field | B = μ₀nI/(2R) | μ₀ = 4π×10⁻⁷ T·m/A |
| Turns | B ∝ n | double n → double B |
| Current | B ∝ I | double I → double B |
| Radius | B ∝ 1/R | double R → half B |
| Solve for radius | R = μ₀nI/(2B) | constant across trials |
Instructions — Running the Virtual Experiment
The Loop & Field Meter tab lets you set the current and number of turns and read the field at the centre; the Field vs Turns tab records B against n so you can confirm the proportionality and recover the radius. Record every reading in your lab report with screenshots.
Simulation — Current-Carrying Loop
Loop settings
Record turns (I = 3 A)
| Turns n | B (mT) | R = μ₀nI/(2B) (m) |
|---|---|---|
| Click a number of turns to record it. | ||
Team Questions
Example Lab Report
Sample report demonstrating the expected format and level of detail. Use as a guide for your own submission, and include labelled screenshots of each loop configuration and field-meter reading.
Magnetism - Magnetic Field in a Current-Carrying Wire Loop
Physics | Section: [Your Section] | Date: [Date]
Lab Members: [Names of all members present]
Purpose
To measure the magnetic field at the centre of a current-carrying wire loop for several numbers of turns, to verify that the field is proportional to the number of turns, and to use B = μ₀nI/(2R) to determine the loop radius — confirming that the calculated radius is consistent across all trials.
Theory
A current-carrying loop produces a magnetic field along its axis, strongest at the centre, given by B = μ₀nI/(2R), where μ₀ = 4π×10⁻⁷ T·m/A. The field is proportional to the number of turns and the current, and inversely proportional to the radius. Rearranged, R = μ₀nI/(2B), so measuring B for a known n and I gives the radius.
μ₀ = 4π×10⁻⁷ T·m/A, I = 3.0 A
Sample Calculation — n = 2
B = μ₀nI/(2R) = (4π×10⁻⁷)(2)(3.0)/(2 × 0.0500) = 7.54×10⁻⁵ T = 0.0754 mT
R = μ₀nI/(2B) = (4π×10⁻⁷)(2)(3.0)/(2 × 7.54×10⁻⁵) = 0.0500 m
Results Table (I = 3.0 A)
| Trial | Turns n | Measured B (mT) | Calculated R (m) |
|---|---|---|---|
| 1 | 1 | 0.0377 | 0.0500 |
| 2 | 2 | 0.0754 | 0.0500 |
| 3 | 3 | 0.1131 | 0.0500 |
| 4 | 5 | 0.1885 | 0.0500 |
| 5 | 10 | 0.3770 | 0.0500 |
Average radius = 0.0500 m. A plot of B against n was a straight line through the origin with slope μ₀I/(2R) = 3.77×10⁻⁵ T/turn.
Discussion
The magnetic field increased in direct proportion to the number of turns: doubling n from 1 to 2 doubled B from 0.0377 to 0.0754 mT, and the n = 10 field was ten times the single-turn value. Solving R = μ₀nI/(2B) gave 0.0500 m for every trial, as expected, because the physical loop did not change — only the number of turns did. The graph of B versus n was linear through the origin, confirming B ∝ n, with a slope equal to μ₀I/(2R).
Small differences between trials would come from the resolution of the field meter and from reading the values, rather than from any real change in the radius. This experiment demonstrates the direct link between electricity and magnetism: an electric current alone, with no magnets present, produces a measurable magnetic field.
Conclusion
The magnetic field at the centre of the loop followed B = μ₀nI/(2R): it was proportional to the number of turns, and the radius recovered from each measurement was a consistent 0.0500 m. The results confirm the fundamental relationship between current and magnetic field.
Practice Questions
Show all work and include units. Use μ₀ = 4π×10⁻⁷ T·m/A and B = μ₀nI/(2R).