To Measure Magnetic Field along the Axis of a Current-Carrying Circular Coil
1. Aim
To measure the magnetic field strength along the axis of a current-carrying circular coil and verify the relationship between the magnetic field and distance from the center of the coil.
2. Apparatus Used
- Circular coil of copper wire (known radius and number of turns)
- Hall effect sensor or magnetic field probe
- DC power supply (0-5A)
- Ammeter (0-5A)
- Connecting wires
- Rheostat (0-10Ω)
- Key/Switch
- Meter scale or traveling microscope
- Stand and clamps for mounting apparatus
- Gauss meter (if Hall effect sensor is not available)
3. Diagram
4. Theory
When a current flows through a circular coil, it produces a magnetic field. The magnetic field at any point on the axis of the coil can be calculated using the Biot-Savart law.
Consider a circular coil of radius $R$ with $N$ turns carrying a current $I$. According to the Biot-Savart law, the magnetic field $B$ at a point $P$ located at a distance $x$ from the center of the coil along its axis is given by:
Where:
- $B$ is the magnetic field in tesla (T)
- $\mu_0$ is the permeability of free space ($4\pi \times 10^{-7}$ H/m)
- $N$ is the number of turns in the coil
- $I$ is the current flowing through the coil in amperes (A)
- $R$ is the radius of the coil in meters (m)
- $x$ is the distance from the center of the coil along its axis in meters (m)
At the center of the coil (when $x = 0$), the magnetic field is:
This experiment aims to verify this theoretical relationship by measuring the magnetic field at various points along the axis of the coil and comparing it with the calculated values.
5. Formula
The key formulas used in this experiment are:
1. Magnetic field at a point on the axis at distance $x$ from the center:
$B = \frac{\mu_0 N I R^2}{2(R^2 + x^2)^{3/2}}$2. Magnetic field at the center of the coil ($x = 0$):
$B_0 = \frac{\mu_0 N I}{2R}$3. Permeability of free space:
$\mu_0 = 4\pi \times 10^{-7}$ H/m6. Procedure
- Measure and record the radius $R$ of the circular coil and count the number of turns $N$.
- Set up the apparatus as shown in the diagram, with the circular coil mounted on a stand.
- Connect the circuit with the DC power supply, ammeter, rheostat, and switch in series with the coil.
- Mount the Hall effect sensor or magnetic field probe on a stand such that it can be moved along the axis of the coil. Ensure that the sensor is properly calibrated.
- Close the switch and adjust the rheostat to set a suitable current (say 1A) in the circuit.
- Place the Hall probe at the center of the coil (i.e., $x = 0$) and record the magnetic field reading.
- Now move the probe along the axis of the coil in small increments (e.g., 1 cm), measuring and recording the magnetic field at each position.
- Take readings both in the positive and negative directions along the axis (i.e., on both sides of the coil).
- Repeat the experiment for at least two more current values (e.g., 2A and 3A).
- Switch off the power supply and disconnect the circuit.
- Calculate the theoretical values of the magnetic field at each position using the formula and compare with the experimental values.
7. Observation Table
Radius of the coil, $R$ = _______ cm
Number of turns, $N$ = _______
| S.No. | Current $I$ (A) | Distance from center $x$ (cm) | Magnetic Field $B$ (mT) | Percentage Error $\left|\frac{B_{exp} - B_{theo}}{B_{theo}}\right| \times 100\%$ |
|
|---|---|---|---|---|---|
| Experimental $B_{exp}$ | Theoretical $B_{theo}$ | ||||
| 1 | |||||
| 2 | |||||
| 3 | |||||
| 4 | |||||
| 5 | |||||
| 6 | |||||
| 7 | |||||
| 8 | |||||
| 9 | |||||
| 10 | |||||
8. Calculations
For each measurement, calculate the theoretical value of the magnetic field using the formula:
Where:
- $\mu_0 = 4\pi \times 10^{-7}$ H/m
- $N$ = Number of turns in the coil
- $I$ = Current in amperes
- $R$ = Radius of the coil in meters
- $x$ = Distance from the center in meters
Calculate the percentage error for each measurement using:
Sample Calculation:
For a set of values, show the detailed calculation:
Given:
- $N$ = (Number of turns)
- $R$ = (Radius in meters)
- $I$ = (Current in amperes)
- $x$ = (Distance in meters)
Step 1: Calculate the theoretical magnetic field
Step 2: Compare with the experimental value and calculate the percentage error
9. Result
- The magnetic field strength $B$ along the axis of a current-carrying circular coil was measured at various distances from the center.
- The experimental values were compared with theoretical values calculated using the Biot-Savart law.
- The average percentage error in the measurements was found to be _______%. (To be filled after experiment)
- The relationship between magnetic field strength and distance from the center was found to follow the theoretical prediction where $B \propto \frac{1}{(R^2 + x^2)^{3/2}}$.
- A graph of $B$ vs $x$ shows the characteristic bell-shaped curve as predicted by theory.
- The magnetic field is maximum at the center of the coil and decreases with increasing distance from the center.
10. Precautions
- Ensure that the Hall effect sensor or magnetic field probe is properly calibrated before use.
- The coil should be perfectly circular for accurate results.
- Ensure that the Hall probe is aligned exactly along the axis of the coil.
- Avoid using high currents for extended periods to prevent heating of the coil, which may alter its resistance.
- Keep magnetic materials and other current-carrying conductors away from the setup to avoid interference.
- Measure the distance from the center of the coil accurately.
- Ensure that the ammeter readings are stable when taking measurements.
- Take multiple readings at each position and use the average value to minimize random errors.
- Switch off the power supply when not taking readings to avoid unnecessary heating of the coil.
- Ensure that the coil is securely mounted and does not move during the experiment.
11. Viva Voice Questions
Q1: What is the Biot-Savart law and how is it applied in this experiment?
The Biot-Savart law relates the magnetic field to the magnitude, direction, length, and proximity of the electric current. In this experiment, it's applied to calculate the magnetic field along the axis of a circular current-carrying coil. The integrated form of the law gives us the formula $B = \frac{\mu_0 N I R^2}{2(R^2 + x^2)^{3/2}}$ for the magnetic field at a distance $x$ from the center of the coil.
Q2: Why does the magnetic field decrease as we move away from the center of the coil along its axis?
As we move away from the center of the coil along its axis, the magnetic field decreases because the distance between the current element and the point of observation increases. According to the Biot-Savart law, the magnetic field is inversely proportional to the square of the distance. Additionally, the effective component of the magnetic field contributing to the axial direction decreases with distance.
Q3: At what point along the axis is the magnetic field maximum?
The magnetic field is maximum at the center of the coil (when $x = 0$). This is because at this point, all the magnetic field contributions from different parts of the coil are optimally aligned and add up constructively.
Q4: How does the magnetic field vary with current?
The magnetic field is directly proportional to the current flowing through the coil. If the current is doubled, the magnetic field also doubles, assuming all other parameters remain constant.
Q5: What are the possible sources of error in this experiment?
Possible sources of error include: misalignment of the Hall probe from the axis, non-uniformity in the coil shape, variation in current due to heating of the coil, calibration errors in the magnetic field sensor, measurement errors in determining distances, and external magnetic interference.
Q6: How would the magnetic field change if the radius of the coil is increased but the current remains the same?
If the radius of the coil is increased while keeping the current constant, the magnetic field at the center would decrease since $B_0 = \frac{\mu_0 N I}{2R}$. The magnetic field is inversely proportional to the radius at the center. However, at points away from the center, the relationship is more complex and depends on the specific distance from the center relative to the radius.
Q7: How does the number of turns in the coil affect the magnetic field?
The magnetic field is directly proportional to the number of turns in the coil. If the number of turns is doubled, the magnetic field also doubles, assuming all other parameters remain constant.
Q8: What is the direction of the magnetic field along the axis of the circular coil?
The direction of the magnetic field along the axis of a circular coil is parallel to the axis. Using the right-hand rule, if the current flows in a counterclockwise direction when viewed from the positive x-axis, the magnetic field points in the positive x-direction.
Q9: How would you modify this experiment to measure the magnetic field at points not on the axis of the coil?
To measure the magnetic field at points not on the axis, I would mount the Hall probe on a two-dimensional traversing mechanism that allows movement in both the axial and radial directions. The formula for calculating the magnetic field would also need to be modified to account for the off-axis position using the more general form of the Biot-Savart law.
Q10: What is the significance of the permeability of free space ($\mu_0$) in this experiment?
The permeability of free space ($\mu_0$) is a fundamental physical constant that appears in the formulation of the Biot-Savart law and other electromagnetic equations. It represents how easily a magnetic field can be established in vacuum and has a value of $4\pi \times 10^{-7}$ H/m. It serves as a proportionality constant in the relationship between current and magnetic field.