Physical Principles of Torque on a Current-Carrying Coil
When an electric current flows through a coil placed in a magnetic field, each segment of the coil carrying current experiences a magnetic force due to the interaction between the magnetic field and the moving charges.
These forces act on opposite sides of the coil in opposite directions, creating a couple that produces a rotational force or torque on the coil.
This torque tends to rotate the coil about its axis, which is the fundamental principle behind devices such as the d.c. motor.
- Magnetic force on a current-carrying conductor: F = BIL sin θ, where B is magnetic flux density (teslas, T), I is current (amperes, A), L is length of conductor in the field (metres, m), and θ is the angle between conductor and magnetic field.
- For a rectangular coil, forces on opposite sides produce a turning effect (torque).
- Torque causes rotational motion of the coil in the magnetic field.
Factors Affecting the Rotational Force (Torque)
The magnitude of the torque (τ) on a coil in a magnetic field depends on three main factors: the number of turns in the coil (N), the current flowing through the coil (I), and the magnetic field strength (B).
Increasing any of these factors increases the torque, making the coil rotate more strongly.
- Number of turns (N): More turns mean more conductors experience force, increasing total torque.
- Current (I): Higher current increases the magnetic force on each conductor, increasing torque.
- Magnetic field strength (B): Stronger magnetic fields exert greater force on the current-carrying conductors, increasing torque.
Mathematical Expression for Torque on a Coil
The torque τ on a rectangular coil of N turns, each of length L and width w, carrying current I in a uniform magnetic field B is given by:
τ = N × B × I × A × sin θ, where A = L × w is the area of the coil and θ is the angle between the coil's plane and the magnetic field direction.
When the coil is perpendicular to the magnetic field (θ = 90°), sin θ = 1 and torque is maximum: τ_max = N B I A.
- Torque (τ) is measured in newton-metres (N·m).
- Area (A) is in square metres (m²).
- Magnetic flux density (B) is in teslas (T).
- Current (I) is in amperes (A).
- Number of turns (N) is dimensionless.
Diagram Construction for a Current-Carrying Coil in a Magnetic Field
To draw a diagram illustrating the rotational force on a coil in a magnetic field:
1. Draw a rectangular coil with several turns (parallel lines close together) on a plane.
2. Indicate the direction of current in the coil using arrows along the wire segments (e.g., clockwise or anticlockwise). 3. Draw magnetic field lines as parallel arrows passing through the coil, labelled B, showing direction from north to south poles of magnets placed on either side of the coil (e.g., left to right). 4. Show forces acting on opposite sides of the coil with arrows perpendicular to the wire segments and magnetic field, using Fleming’s left-hand rule to determine force direction. 5. Indicate the torque direction with a curved arrow around the coil’s axis of rotation.
- Use Fleming’s left-hand rule: thumb = force (motion), first finger = magnetic field, second finger = current.
- Opposite forces on opposite sides create a turning effect (torque).
Worked Example: Calculating Torque on a Coil
Calculate the torque on a rectangular coil with 50 turns, each of area 0.02 m², carrying a current of 3.0 A in a magnetic field of strength 0.4 T. The coil is perpendicular to the magnetic field.
Solution: Use the formula τ = N B I A sin θ. Since coil is perpendicular, sin 90° = 1.
Substitute values: τ = 50 × 0.4 T × 3.0 A × 0.02 m² × 1 = 1.2 N·m.
- Torque τ = 1.2 newton-metres.
- Common error: forgetting to multiply by number of turns or using wrong units.
At-a-glance summary
Comparison table
| Factor | Physics Detail | Effect on Torque | Exam Use |
|---|---|---|---|
| Number of turns (N) | More turns mean more current-carrying conductors in the magnetic field | Torque increases proportionally with N | State that increasing coil turns increases torque |
| Electric current (I) | Current produces magnetic force on conductors | Torque increases proportionally with I | Explain how increasing current increases torque |
| Magnetic field strength (B) | Stronger magnetic field exerts greater force on current | Torque increases proportionally with B | Describe effect of stronger magnetic field on torque |
| Coil area (A) | Larger coil area means longer wire in field and larger moment arm | Torque increases with coil area | Use coil dimensions to calculate torque |
Active recall
Quick check
1. What is the effect of increasing the current in a coil placed in a magnetic field?
Increasing the current increases the magnetic force on the coil, thus increasing the rotational force (torque).
2. How does increasing the number of turns in the coil affect the torque?
Increasing the number of turns increases the total force on the coil, increasing the torque proportionally.
3. Write the formula for the torque on a coil in a magnetic field and define each symbol.
τ = N B I A sin θ, where τ is torque (N·m), N is number of turns (dimensionless), B is magnetic flux density (T), I is current (A), A is coil area (m²), and θ is angle between coil plane and magnetic field.
4. If the magnetic field strength doubles, what happens to the torque on the coil?
The torque doubles because torque is directly proportional to the magnetic field strength B.