Force And Torque Due To Magnetic Field
- Torque on a current carrying loop placed in magnetic field
- Magnetic moment
Introduction to Force and Torque
- When a current-carrying loop is placed in a magnetic field, it experiences both force and torque.
- The force and torque can be determined using the concept of magnetic moment.
- In this section, we will discuss the torque on a current carrying loop placed in a magnetic field.
Torque on a current carrying loop
- A loop carrying a current (I) is placed in a magnetic field (B).
- The magnetic field exerts a force on each segment of the loop.
- The net effect of these forces is a torque on the loop.
- The torque can be calculated using the formula:
- τ = NIA x B
- where τ is the torque,
- N is the number of turns in the loop,
- A is the area of the loop,
- B is the magnetic field,
- x represents the cross product.
- τ = NIA x B
Magnetic Moment
- The magnetic moment (μ) of a current-carrying loop is a vector quantity that represents the strength and orientation of the magnetic field produced by the loop.
- It is defined as the product of the current (I) and the area (A) vector of the loop:
- μ = I x A
- where μ is the magnetic moment,
- I is the current,
- A is the area vector.
- μ = I x A
Relation Between Torque and Magnetic Moment
- The torque (τ) exerted on a loop by a magnetic field can be related to the magnetic moment (μ) of the loop.
- The torque is given by the formula:
- τ = μ x B
- where τ is the torque,
- μ is the magnetic moment,
- B is the magnetic field,
- x represents the cross product.
- τ = μ x B
- This equation shows that the torque is directly proportional to the magnetic moment and the magnetic field.
Example: Torque Calculation
- Let’s consider a rectangular loop with dimensions 2 cm x 3 cm, carrying a current of 2 A.
- The loop is placed in a magnetic field of magnitude 0.5 T.
- To calculate the torque on the loop, we can use the formula τ = μ x B.
- First, we need to calculate the magnetic moment (μ) of the loop.
- The area (A) of the loop can be calculated as A = l x w = 0.02 m x 0.03 m = 0.0006 m^2.
- Therefore, the magnetic moment is given by μ = I x A = 2 A x 0.0006 m^2 = 0.0012 A m^2.
- Finally, we can calculate the torque as τ = μ x B = 0.0012 A m^2 x 0.5 T = 0.0006 N m.
- Hence, the torque exerted on the loop is 0.0006 N m.
Summary
- When a current-carrying loop is placed in a magnetic field, it experiences both force and torque.
- The torque on the loop can be calculated using the formula τ = NIA x B.
- The magnetic moment of a current-carrying loop is given by the formula μ = I x A.
- The torque exerted on the loop is related to the magnetic moment and the magnetic field through the equation τ = μ x B.
- The torque is directly proportional to the magnetic moment and the magnetic field.
Key Points to Remember
- When a current-carrying loop is placed in a magnetic field, it experiences both force and torque.
- The torque on the loop can be calculated using the formula τ = NIA x B.
- The magnetic moment of a current-carrying loop is given by the formula μ = I x A.
- The torque exerted on the loop is related to the magnetic moment and the magnetic field through the equation τ = μ x B.
- The torque is directly proportional to the magnetic moment and the magnetic field.
References
- Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of Physics (10th ed.). John Wiley & Sons.
- Pervez, A. (n.d.). Magnetic Torque on a Current Loop. Retrieved from https://www.cleariitmedical.com/2020/12/magnetic-torque-current-loop-can-you-cut-mathematics.html
Torque on a current carrying loop - continued
The torque on a current-carrying loop can also be expressed in terms of the magnetic moment:
- τ = μ x B
- where τ is the torque,
- μ is the magnetic moment,
- B is the magnetic field,
- x represents the cross product.
Calculation of Torque using Magnetic Moment
To calculate the torque on a current-carrying loop using the magnetic moment, you can use the formula:
- τ = μ x B For example, consider a circular loop with a diameter of 0.1 m and a current of 5 A. The loop is placed in a magnetic field of magnitude 0.3 T. Calculate the torque on the loop.
Calculation of Torque using Magnetic Moment - continued
First, calculate the area of the loop:
- A = π r^2 = π (0.05 m)^2 = 0.00785 m^2 Then, calculate the magnetic moment:
- μ = I x A = 5 A x 0.00785 m^2 = 0.03925 A.m^2 Finally, calculate the torque using the formula:
- τ = μ x B = 0.03925 A.m^2 x 0.3 T = 0.011775 N.m
Torque on a current carrying loop - Summary
To summarize:
- The torque on a current-carrying loop can be calculated using the magnetic moment and the magnetic field: τ = μ x B
- The magnetic moment of a loop is given by μ = I x A, where I is the current and A is the area of the loop.
- The torque is directly proportional to the magnetic moment and the magnetic field.
Example: Application of Torque on a Loop
An electric motor consists of a rectangular loop of wire carrying a current of 2 A. The loop has dimensions 0.1 m x 0.2 m and is placed in a magnetic field of 0.5 T. Calculate the torque on the loop.
Example: Application of Torque on a Loop - continued
First, calculate the area of the loop:
- A = l x w = 0.1 m x 0.2 m = 0.02 m^2 Then, calculate the magnetic moment:
- μ = I x A = 2 A x 0.02 m^2 = 0.04 A.m^2 Finally, calculate the torque using the formula:
- τ = μ x B = 0.04 A.m^2 x 0.5 T = 0.02 N.m
Torque and Rotational Equilibrium
If the torque on a current-carrying loop is zero, the loop will be in rotational equilibrium. In other words, the loop will not rotate. For a loop to be in rotational equilibrium, the torque applied to it must be balanced by an equal and opposite torque.
Application of Torque on a Loop - Electric Motors
Electric motors use the torque on a current-carrying loop to convert electrical energy to mechanical energy. The torque exerted on the loop causes it to rotate, resulting in the movement of various mechanical components. This rotational motion is used to perform tasks such as spinning a fan, rotating a shaft, or powering a vehicle.
Application of Torque on a Loop - Electric Motors - continued
The direction of rotation of the loop depends on the direction of the current and the orientation of the loop in the magnetic field.
You can change the direction of rotation by reversing the direction of the current or by changing the orientation of the loop.
Summary
To summarize, in this section we discussed:
- The torque on a current-carrying loop can be calculated using the magnetic moment and the magnetic field.
- The torque is directly proportional to the magnetic moment and the magnetic field.
- If the torque on a loop is zero, it will be in rotational equilibrium.
- Electric motors use the torque on a current-carrying loop to convert electrical energy to mechanical energy.
Ampere’s Law
- Ampere’s law relates the magnetic field around a closed loop to the electric current passing through the loop.
- It states that the integral of the magnetic field along a closed loop is equal to the product of the current enclosed by the loop and a constant called the permeability of free space (μ₀).
- Mathematically, it can be written as:
- ∮ B · dl = μ₀ I_enclosed
Applications of Ampere’s Law
Ampere’s law is useful in many applications of magnetism, including:
- Calculating the magnetic field inside a long straight conductor.
- Determining the magnetic field around a current-carrying wire.
- Calculating the magnetic field inside a solenoid.
Magnetic Field Inside a Long Straight Conductor
- Ampere’s law can be used to calculate the magnetic field inside a long straight conductor carrying a current I.
- The magnetic field inside the conductor is uniform and its magnitude is given by:
- B = (μ₀ I) / (2π r)
- where B is the magnetic field,
- μ₀ is the permeability of free space,
- I is the current,
- r is the distance from the center of the conductor.
- B = (μ₀ I) / (2π r)
Magnetic Field Around a Current-Carrying Wire
- Ampere’s law can also be used to determine the magnetic field around a current-carrying wire.
- The magnetic field follows concentric circles around the wire, with its magnitude given by:
- B = (μ₀ I) / (2π r)
- where B is the magnetic field,
- μ₀ is the permeability of free space,
- I is the current,
- r is the distance from the wire.
- B = (μ₀ I) / (2π r)
Magnetic Field Inside a Solenoid
- A solenoid is a long coil of wire wound closely in the form of a cylinder.
- Ampere’s law can be used to calculate the magnetic field inside a solenoid.
- Inside the solenoid, the magnetic field is uniform and its magnitude is given by:
- B = μ₀ n I
- where B is the magnetic field,
- μ₀ is the permeability of free space,
- n is the number of turns per unit length (also called the coil density),
- I is the current.
- B = μ₀ n I
Magnetic Force on a Moving Charge
- A moving charge experiences a force in the presence of a magnetic field.
- The magnitude of the magnetic force (F) on a charge (q) moving with a velocity (v) in a magnetic field (B) is given by the formula:
- F = q v B sin(θ)
- where θ is the angle between the velocity vector and the magnetic field vector.
- F = q v B sin(θ)
Magnetic Force on a Current-Carrying Wire
- A current-carrying wire also experiences a force when placed in a magnetic field.
- The magnitude of the magnetic force (F) on a length (L) of wire carrying a current (I) in a magnetic field (B) is given by the formula:
- F = I L B sin(θ)
- where θ is the angle between the wire and the magnetic field vector.
- F = I L B sin(θ)
Magnetic Force on a Current-Carrying Wire - Example
- For example, consider a wire of length 0.2 m carrying a current of 4 A placed in a magnetic field of magnitude 0.5 T.
- If the angle between the wire and the magnetic field is 30 degrees, calculate the magnitude of the magnetic force on the wire.
- Using the formula F = I L B sin(θ):
- F = (4 A) x (0.2 m) x (0.5 T) x sin(30 degrees)
- F ≈ 0.2 N
Magnetic Force on a Current-Carrying Wire - Continued
- The direction of the magnetic force on a current-carrying wire can be determined using the right-hand rule.
- If the thumb of the right hand points in the direction of the current and the fingers point in the direction of the magnetic field, then the palm of the hand will face the direction of the magnetic force.
Summary
To summarize, in this section we discussed:
- Ampere’s law and its applications in calculating magnetic fields.
- The magnetic field inside a long straight conductor, around a current-carrying wire, and inside a solenoid.
- The magnetic force on a moving charge and a current-carrying wire.
- How to calculate the magnitude and direction of the magnetic force using appropriate formulas and the right-hand rule.
