How to Calculate the Magnetic Field on a Circular Loop’s Axis
The magnetic field on the axis of a circular current loop describes the magnitude and direction of the magnetic field generated at any point along the symmetry axis passing through the center of a circular wire carrying electric current. This concept is essential for understanding electromagnetic effects in devices such as coils, galvanometers, and electromagnets.
Biot–Savart Law and Its Application to a Circular Current Loop
The Biot–Savart law provides the fundamental relationship used to calculate the magnetic field generated by a small segment of a current-carrying conductor. For a current element $I d\vec{l}$ at a position relative to a point $P$ at distance $r$, the law states:
$\displaystyle d\vec{B} = \dfrac{\mu_0}{4\pi} \dfrac{I\, d\vec{l} \times \hat{r}}{r^2}$
In the case of a circular loop of radius $a$ carrying current $I$, the field is calculated at a point $P$ located at a distance $x$ along the axis passing perpendicularly through the center of the loop. The symmetry of the problem allows the resultant magnetic field to be determined by integrating contributions from all current elements.
Expression for Magnetic Field on the Axis of a Circular Loop
Consider a circular loop of radius $a$ centered at $O$ on the $xy$-plane, carrying current $I$. Let $P$ be a point on the axis of the loop at distance $x$ from $O$. The magnetic field due to a current element $d\vec{l}$ at $P$ is given by the component along the axis:
$\displaystyle dB = \dfrac{\mu_0}{4\pi} \dfrac{I\, dl \sin\theta}{(a^2 + x^2)}$
Here, $\theta$ is the angle between the position vector of $P$ with respect to the element and the axis. The net perpendicular components of $dB$ from diametrically opposite elements cancel due to symmetry, and only the axial components add constructively. The axial component is $dB \cos\theta$.
For geometrical relation, $\sin\theta = \dfrac{a}{\sqrt{a^2 + x^2}}$ and $\cos\theta = \dfrac{x}{\sqrt{a^2 + x^2}}$.
So, the axial component at $P$ contributed by the element is:
$dB_{\text{axis}} = dB \cos\theta = \dfrac{\mu_0}{4\pi} \dfrac{I\, dl \cdot a}{(a^2 + x^2)^{3/2}}$
Integrating $dl$ over the whole circumference ($2\pi a$):
$B = \dfrac{\mu_0 I a^2}{2(a^2 + x^2)^{3/2}}$
For a coil of $N$ turns, the result is multiplied by $N$, giving:
$B = \dfrac{\mu_0 N I a^2}{2(a^2 + x^2)^{3/2}}$
Key Parameters in the Magnetic Field Formula
The terms in the expression for the magnetic field are explained in the following table for clarity. This aids in the correct application of the formula in numerical and theoretical questions.
| Symbol | Meaning |
|---|---|
| $\mu_0$ | Permeability of free space ($4\pi \times 10^{-7}$ H/m) |
| $I$ | Current in the loop (in A) |
| $a$ | Radius of the loop (in m) |
| $x$ | Axial distance from center (in m) |
| $N$ | Number of turns in the coil |
| $B$ | Magnetic field on the axis (in Tesla) |
Special Case: Magnetic Field at the Center of the Loop
At the center of the loop ($x = 0$), the expression simplifies significantly. Substituting $x = 0$ yields:
$B_{\text{center}} = \dfrac{\mu_0 N I}{2a}$
This is the standard formula for the magnetic field at the geometric center of a circular current loop. This value is important for experimental setups requiring uniform magnetic fields at the center.
Direction of the Magnetic Field on the Axis
The direction of the magnetic field on the axis of a circular loop is determined using the right-hand thumb rule. If the fingers of the right hand curl in the direction of the current, the thumb points along the direction of the axial magnetic field. This directional property is essential when analyzing the resultant field in coil systems.
Variation of the Magnetic Field with Distance and Radius
The magnetic field strength on the axis of a circular loop varies with distance $x$ and radius $a$. As $x$ increases, the denominator $(a^2 + x^2)^{3/2}$ increases, causing the field to decrease rapidly. For a fixed current, increasing the radius increases the field at the center but decreases it at points far from the loop. These relationships are necessary in the design of electromagnetic devices.
Applications of the Axial Magnetic Field of a Circular Loop
Circular current loops form the basis of many electromagnetic devices. The principle is used in the design of electromagnets, MRI scanner magnets, moving coil galvanometers, and induction coils. Understanding the field along the axis is also fundamental to experimental arrangements studying electromagnetic induction and related topics in Electromagnetic Induction and AC Revision Notes.
- Basis for electromagnetic coil design
- Used in sensitive measuring instruments
- Applied in medical imaging devices
- Essential for magnetic field control setups
Important Reminders for Calculations in JEE Main
It is important to distinguish between the field at the center of the loop and at an arbitrary point on its axis. The denominator in the formula must be cubed, not just square rooted. The number of turns $N$ must be included when the coil has multiple turns. The direction should always be checked using the right-hand thumb rule. For related concepts, refer to Magnetic Effects of Current and Magnetism.
Key Formulae for Quick Revision
The following formulae should be memorized for use in JEE Main problems:
- On the axis: $B = \dfrac{\mu_0 N I a^2}{2(a^2 + x^2)^{3/2}}$
- At the center: $B = \dfrac{\mu_0 N I}{2a}$
Magnetic Field Visualization and Field Lines
The pattern of magnetic field lines produced by a circular loop carrying current is concentric near the wire and more uniform along the axis, especially near the center. The direction and shape of these field lines can be visualized or simulated in three dimensions for conceptual clarity. For more on magnetic field lines, see Magnetic Lines of Force.
Comparison with Other Current Configurations
The expression for the axial magnetic field of a circular loop differs from those for straight wires or solenoids. While Ampère’s Law is suitable for straight wires, the Biot–Savart law must be used for rings due to the absence of sufficient symmetry. Further, the field at the center of a circular loop is distinct from that at any point on its axis. For other configurations, refer to Magnetic Field Due to Infinite Wire and Magnetic Field Due to Straight Wire.
Summary Table: Magnetic Field on the Axis of a Circular Loop
| Case | Magnetic Field ($B$) |
|---|---|
| At axial point ($x$) | $\dfrac{\mu_0 N I a^2}{2(a^2 + x^2)^{3/2}}$ |
| At center ($x = 0$) | $\dfrac{\mu_0 N I}{2a}$ |
Practical Considerations and Problem Solving Tips
When applying the formula, ensure dimensional consistency and include the correct value for the permeability of free space. Pay attention to the number of turns and the case (center vs. axis). For further understanding of concepts such as magnetic moment, refer to Understanding Magnetic Moment.
FAQs on Magnetic Field on the Axis of a Circular Current Loop
1. What is the magnetic field on the axis of a circular current loop?
The magnetic field on the axis of a circular current loop is given by a specific formula that depends on the current, radius of the loop, and the distance from the center.
• The field at a point on the axis at a distance x from the center is:
B = (μ₀ I R²) / [2(R² + x²)^{3/2}]
where μ₀ is the permeability of free space, I is the current, R is the radius, and x is the distance from center along the axis.
• This key formula is used in physics to determine magnetic field strength for circular loops at various positions.
2. How do you derive the expression for magnetic field on the axis of a circular current loop?
The magnetic field on the axis of a current loop is derived using the Biot-Savart Law.
• Consider a current I flowing in a circular loop of radius R at the origin.
• Select a point P on the axis, distance x from the center.
• Integrate the contributions due to all elements (dl) around the loop using the Biot-Savart Law.
• The only component that survives is along the axis due to symmetry.
• The final result is:
B = (μ₀ I R²) / [2(R² + x²)^{3/2}]
3. What factors affect the magnetic field at the axis of a circular loop?
The magnetic field at the axis of a circular current loop is influenced by several factors:
• The magnitude of the current (I) flowing through the loop.
• The radius (R) of the loop.
• The distance (x) from the center along the axis.
• The number of turns (n) if it is a coil, the field multiplies by n.
• The permeability of free space (μ₀).
All these parameters collectively determine the field using the standard formula.
4. What is the direction of the magnetic field produced by a circular current loop on its axis?
The direction of the magnetic field on the axis of a circular loop is determined using the right-hand rule:
• Curl your right hand fingers in the direction of current.
• Your thumb points in the direction of the magnetic field on the axis.
• This direction is along the axis passing through the center of the loop.
5. What happens to the magnetic field at the centre of a circular current loop?
At the centre of a circular current loop (x = 0), the expression simplifies:
• Field at center: B = (μ₀ I) / (2R)
• The field is maximum at the center for a given current and radius.
• It points perpendicular to the plane of the loop as given by the right-hand rule.
6. Why does the magnetic field decrease as we move away from the center along the axis of a current loop?
The magnetic field decreases with distance from the center because the denominator of the formula increases faster than the numerator:
• As x increases, (R² + x²)^{3/2} grows, reducing the field value.
• The spreading out of field lines along the axis causes a reduction in field strength.
• For points far from the loop (x ≫ R), the field falls off as 1/x³ (like a magnetic dipole).
7. What is the use of a circular current loop in practical applications?
Circular current loops are used in many practical devices to create controlled magnetic fields:
• Galvanometers and moving coil instruments.
• Electromagnets and MRI machines.
• Loudspeakers.
• Particle accelerators.
They are fundamental for generating uniform and predictable magnetic fields along a given direction.
8. Does the magnetic field on the axis depend on the shape of the loop?
Yes, the shape of the current-carrying conductor affects the magnetic field.
• The standard formula applies specifically to circular loops.
• Other shapes (square, rectangular) have different field distributions and require separate analysis.
• The circular loop produces a symmetric field, making calculations simpler and fields stronger at the centre compared to other shapes.
9. How does the number of turns in the loop affect the magnetic field on its axis?
The magnetic field on the axis is directly proportional to the number of turns (n):
• If the loop is wound n times, field is: B = (μ₀ n I R²) / [2(R² + x²)^{3/2}]
• More turns increase the field strength proportionally at every point along the axis.
10. At what position is the magnetic field maximum on the axis of a circular loop?
The magnetic field is maximum at the center of the circular loop (x = 0).
• At x = 0: B = (μ₀ I) / (2R)
• As you move away from the center, the field decreases symmetrically on both sides along the axis.
11. What is Biot-Savart law and how is it applied to calculate the field of a circular current loop?
Biot-Savart Law explains the magnetic field due to a small current element:
• States that dB = (μ₀/4π) × (I dl × r̂) / r², where dl is a current element and r is the distance to the point.
• For a circular loop, sum the field contributions from all such elements by integration, considering symmetry.
• This gives the axis field formula used in exams.
12. What is the physical significance of the magnetic field produced by a current-carrying circular loop?
The magnetic field of a current-carrying circular loop demonstrates how moving charges produce magnetic effects.
• It helps to explain electromagnetic devices, Earth's magnetism, and magnetic effects in circuits.
• Understanding this field is fundamental for physics, electronics, and engineering problems in the syllabus.
