Calculating Torque on a Rectangular Coil in a Magnetic Field

This problem involves finding the torque experienced by a current-carrying rectangular coil placed within a uniform magnetic field. The torque on such a coil depends on the number of turns, the current, the area of the coil, the magnetic field strength, and the angle between the coil's orientation and the magnetic field.

Torque is a twisting force that tends to cause rotation. In the case of a current coil in a magnetic field, this torque arises from the forces exerted by the magnetic field on the current-carrying wires.

Identifying Physics Parameters

Let's identify the given values from the problem statement:

• Number of turns, \(N = 300\)
• Length of the coil, \(l = 20 \, \text{cm}\)
• Breadth of the coil, \(b = 12 \, \text{cm}\)
• Current in the coil, \(I = 12 \, \text{A}\)
• Magnetic field strength, \(B = 6 \, \text{T}\)
• Angle the plane of the coil makes with the magnetic field, \(\theta' = 60^\circ\)

To use consistent SI units in our calculation, we need to convert the dimensions of the coil from centimeters to meters:

• \(l = 20 \, \text{cm} = 0.20 \, \text{m}\)
• \(b = 12 \, \text{cm} = 0.12 \, \text{m}\)

Determining the Area of the Coil

The area \(A\) of a rectangular coil is calculated by multiplying its length and breadth.

$$A = l \times b$$

Substituting the values in meters:

$$A = 0.20 \, \text{m} \times 0.12 \, \text{m}$$

$$A = 0.024 \, \text{m}^2$$

Understanding Magnetic Torque Formula

The magnitude of the torque \(\tau\) acting on a current loop placed in a magnetic field is given by the formula:

$$\tau = NIAB \sin(\theta)$$

Where:

• \(N\) is the number of turns in the coil.
• \(I\) is the current flowing through the coil.
• \(A\) is the area of the coil.
• \(B\) is the magnetic field strength.
• \(\theta\) is the angle between the magnetic field vector (\(\vec{B}\)) and the normal vector (\(\vec{A}\)) to the plane of the coil.

The normal vector to the plane of the coil is perpendicular to the plane. If the plane of the coil makes an angle \(\theta'\) with the magnetic field, then the angle \(\theta\) between the normal to the plane and the magnetic field is typically given by \(\theta = 90^\circ - \theta'\). In this problem, \(\theta' = 60^\circ\), so the standard angle to use in the sine function would be \(\theta = 90^\circ - 60^\circ = 30^\circ\).

Calculating the Torque Value

Let's calculate the product \(NIAB\), which represents the maximum possible torque the coil could experience in this magnetic field (when \(\sin(\theta)=1\)).

$$NIAB = 300 \times 12 \, \text{A} \times 0.024 \, \text{m}^2 \times 6 \, \text{T}$$

We can multiply these values step by step:

$$NIAB = (300 \times 12) \times 0.024 \times 6$$

$$NIAB = 3600 \times 0.024 \times 6$$

$$NIAB = (3600 \times 0.024) \times 6$$

$$NIAB = 86.4 \times 6$$

$$NIAB = 518.4 \, \text{Nm}$$

Based on the calculation of the coil's parameters (N, I, A, B), the resulting torque value is determined to be 518.4 Nm.

Physics Revision Table: Magnetic Torque Calculation

Additional Information: Magnetic Dipole Moment and Torque Orientation

The torque on a current loop is related to its magnetic dipole moment \(\vec{m}\). For a coil with N turns, the magnitude of the magnetic dipole moment is \(m = NIA\), and its direction is given by the normal vector to the coil's area (following the right-hand rule based on the current direction).

The torque can be expressed in vector form as:

$$\vec{\tau} = \vec{m} \times \vec{B} \quad \text{ (for a single turn)}$$

For N turns, this becomes:

$$\vec{\tau} = N \vec{m}_{loop} \times \vec{B} = (NIA) \hat{n} \times \vec{B}$$

Where \(\hat{n}\) is the unit vector normal to the coil's plane. The magnitude is indeed \(\tau = NIAB \sin(\theta)\), where \(\theta\) is the angle between \(\hat{n}\) (the direction of \(\vec{m}\)) and \(\vec{B}\).

• The torque is maximum when the magnetic dipole moment is perpendicular to the magnetic field (\(\theta = 90^\circ\)). This corresponds to the plane of the coil being parallel to the magnetic field (\(\theta' = 90^\circ\)). The maximum torque is \(\tau_{max} = NIAB\).
• The torque is zero when the magnetic dipole moment is parallel or anti-parallel to the magnetic field (\(\theta = 0^\circ\) or \(180^\circ\)). This corresponds to the plane of the coil being perpendicular to the magnetic field (\(\theta' = 0^\circ\)).

The torque always acts to align the magnetic dipole moment vector \(\vec{m}\) with the magnetic field vector \(\vec{B}\).