Question

\(\mathrm{A}\) bar magnet of length \(10 \mathrm{~cm}\) and having the pole strength equal \(10^{3} \mathrm{Am}\) to is kept in a magnetic field having magnetic induction (B) equal to \(4 \pi \times 10^{3}\) tesla. It makes an angle of \(30^{\circ}\) with the direction of magnetic induction. The value of the torque acting on the magnet is Joule. (a) \(2 \pi \times 10^{-7}\) (b) \(2 \pi \times 10^{5}\) (c) \(0.5\) (d) \(0.5 \times 10^{2}\)

Short answer

The torque acting on the magnet is \(2\pi \times 10^{5} J\).

Step-by-step solution

Identify the formula for torque.

We know that the formula for torque acting on a magnet in a magnetic field is given by: \[\tau = pB \sin{\theta}\] Where: - \(\tau\) is the torque acting on the magnet - \(p\) is the magnetic moment of the magnet - \(B\) is the magnetic induction - \(\theta\) is the angle between the magnet and the magnetic induction direction. Since pole strength and length are given, we need to find the magnetic moment of the magnet first. The magnetic moment is calculated by the product of pole strength and length: \[p = m\times l\]

Find the magnetic moment of the magnet.

We are given the pole strength as \(10^3 Am\) and the length as \(10cm = 0.1m\). Thus, the magnetic moment is: \[p = (10^3 Am) \times (0.1 m) = 100 Am^2\]

Plug the values into the torque formula.

Now that we have the magnetic moment, we can plug the values into the torque formula: \[\tau = pB \sin{\theta} = (100 Am^2)(4\pi\times10^3 T)\sin{30^{\circ}}\]

Calculate the torque.

Now we need to calculate the value of the torque: \[\tau = (100 Am^2)(4\pi\times10^3 T)\sin{30^{\circ}} = 100 \times 4 \pi \times 10^3 \times\frac{1}{2} \] \[\tau = 2\pi \times 10^5Nm\]

Determine the torque in Joules.

We can express the torque value in Joule as well: \[1N.m = 1J\] Therefore, \[\tau = 2\pi \times 10^5J\] The value of the torque acting on the magnet is \(2\pi \times 10^{5} J\). Thus, the correct answer is (b) \(2 \pi \times 10^{5}\).

Key concepts

Magnetic Moment

The magnetic moment is a fundamental concept in understanding the behavior of magnets within a magnetic field. It is a vector quantity that represents the magnetic strength and orientation of a magnet. The formula to determine the magnetic moment \(p\) of a bar magnet is given by the product of its pole strength \(m\) and its length \(l\). This can be expressed as:\[ p = m \times l \]In this problem, the pole strength is provided as \(10^3 \, Am\), and the bar magnet's length is \(0.1 \, m\). Plugging these values into the formula gives:\[ p = (10^3 \, Am) \times (0.1 \, m) = 100 \, Am^2 \]Therefore, the magnetic moment of this particular magnet is \(100 \, Am^2\). This value influences how the magnet interacts with external magnetic fields such as torque.

Magnetic Induction

Magnetic induction, often referred to as magnetic flux density, is an indicator of how strong a magnetic field is at a given point. It is measured in teslas (T). In this exercise, magnetic induction \(B\) is provided as \(4 \pi \times 10^3 \, T\).Magnetic induction is crucial because it affects how a magnetic material like a bar magnet will experience a torque when placed in a magnetic field. This torque is dependent on both the magnetic moment \(p\) and the angle \(\theta\) between the magnetic field and the direction of the magnetic moment. The formula that expresses the relationship of torque \(\tau\) with magnetic induction is:\[ \tau = pB \sin{\theta} \]Where \(\theta\) is the angle which, in this problem, is \(30^{\circ}\). The magnetic induction acts as a multiplier to the magnetic moment to determine the torque experienced by the magnet.

Pole Strength

Pole strength refers to the magnitude of the north or south pole of a magnet, representing how strong the magnet's poles are. It is crucial when calculating the magnetic moment and other magnetic properties. In units of ampere-meter (\(Am\)), it quantifies the magnetic effect of each pole independently.Given in the problem, the pole strength of the magnet is \(10^3 \, Am\). This value is multiplied by the magnet's length to derive the magnetic moment, as follows:- **Formula:** \( p = m \times l \)- **Given Pole Strength (\(m\)):** \( 10^3 \, Am \)- **Length (\(l\)):** \( 0.1 \, m \)Using the values, we computed a magnetic moment \(p\) of \(100 \, Am^2\), which is pivotal for determining other properties like torque when the magnet is placed in a magnetic field. Understanding pole strength helps grasp how powerful the magnet's effect on nearby objects or other fields is.

Magnetic Field

A magnetic field defines the space around a magnetic material or a current-carrying conductor where magnetic forces can be detected. Its presence and orientation affect how objects within it will behave.A key attribute of magnetic fields is magnetic flux density or magnetic induction, denoted by \(B\). In this exercise, the magnetic field is integral to calculating the torque acting on the magnet through the formula:\[ \tau = pB \sin{\theta} \]**Important Points about Magnetic Fields:**- They have both magnitude (given by magnetic induction \(B\)) and direction.- The torque experienced by a magnet depends on the magnetic field and the orientation of the magnet to the field.- This field can be considered uniform or non-uniform depending on the conditions.Given the magnetic induction in this exercise is \(4 \pi \times 10^3 \, T\), it shows that the magnetic field is strong and will markedly influence the magnet's behavior through torque. Magnetic fields help us predict how a magnet will interact in terms of rotation and force with respect to its orientation within that field.