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Chapter 20: Problem 63

A magnetic needle suspended by a silk thread is vibrating in the earth's magnetie field, if the temperature of the needle is increased by \(500^{\circ} \mathrm{C}\), then [BVP Engg. 2006] (a) the time period decreases (b) the time period remains unchanged (c) the time period increases (d) the needle stops vibrating

Short Answer

Expert verified
(c) The time period increases.

Step by step solution

01

Understand the Physical Concept

The motion of a magnetic needle in a magnetic field is related to its moment of inertia and the magnetic field strength. The formula for the time period \(T\) of oscillation is given by \(T = 2\pi \sqrt{\frac{I}{B_p \cdot m}}\), where \(I\) is the moment of inertia, \(B_p\) is the strength of the magnetic field, and \(m\) is the magnetic moment.
02

Analyze the Effect of Temperature

Raising the temperature of the magnetic needle affects its magnetic moment \(m\). As temperature increases, the magnetic moment typically decreases due to thermal agitation aligning the domains in random directions, reducing their alignment along the field.
03

Relate Temperature Change to Time Period

Since the magnetic moment \(m\) decreases with increasing temperature, the denominator in the time period formula \(T = 2\pi \sqrt{\frac{I}{B_p \cdot m}}\) decreases, increasing the time period \(T\). Hence, the time period increases when the needle's temperature is increased.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Moment of Inertia
Moment of inertia is a crucial concept in understanding the motion of a magnetic needle. It describes how much torque is needed for a particular angular acceleration about a rotational axis. Think of it as the rotational equivalent of mass. The formula for the time period of a vibrating magnetic needle is \[ T = 2\pi \sqrt{\frac{I}{B_p \cdot m}} \]where \( I \) (moment of inertia) is a vital component. In this context:
  • Higher moment of inertia means more resistance to changes in motion.
  • Plays a pivotal role in the needle's behavior when subjected to the Earth's magnetic field.
Therefore, understanding how the moment of inertia interacts with other factors helps us comprehend the needle's oscillation.
Magnetic Moment
Magnetic moment is a measure of the strength and direction of a magnet's ability to influence magnetic fields. In our magnetic needle scenario, the magnetic moment \( m \) represents the needle's capacity to interact with the Earth's magnetic field. As the formula for the time period shows:
  • A larger magnetic moment tends to decrease the time period of oscillation.
  • It works with the Earth's magnetic field to stabilize the needle's motion.
Importantly, the magnetic moment is sensitive to changes in environment, like temperature, affecting how the magnetic needle performs in different conditions.
Temperature Effect on Magnetism
Temperature has a profound impact on magnetism. When we increase the temperature of a magnetic needle by 500掳C, as given in the exercise, the magnetic moment is altered. Why does this happen?
  • Thermal energy causes magnetic domains to misalign.
  • This misalignment reduces overall magnetic strength.
As a result, the magnetic moment decreases, leading to an increase in the time period of oscillation when temperature rises, according to the formula. This understanding is key to predicting how a magnetic needle will behave under different thermal conditions.
Earth's Magnetic Field
The Earth's magnetic field is a constant force surrounding us. It's crucial in the functioning of a magnetic needle, as it provides the external magnetic influence needed for oscillation. To comprehend its role in the needle's motion:
  • The Earth's field exerts a stabilizing influence, allowing the needle to align itself along its lines.
  • It combines with the magnetic moment to determine oscillation characteristics.
By understanding this constant magnetic presence, we gain insight into how the needle reacts when external conditions, like temperature, are varied.

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Most popular questions from this chapter

Magnetic field of earth is identical to magnetic field of a giant magnet held \(20^{\circ}\) west of geographic \(N-\mathrm{S}\) at the centre of earth. At equator, horizontal component of earth is 0.32 G. Vertical component can be calculated from the relation \(V=H \tan \delta\), where \(\delta\) is angle of dip at the place. The value of \(\delta=0^{\circ}\) at equator and \(8=90^{\circ}\) at poles. Magnetie field on the surface of earth is of the order of (a) \(10^{-4} \mathrm{~T}\) (b) \(10^{-4} \mathrm{G}\) (c) \(10^{-5} \mathrm{~T}\) (d) \(10^{-5} \mathrm{G}\)

A long solenoid has 1000 turs per metre and carries a current of \(1 \mathrm{~A}\). It has a soft iron core of \(\mu_{r}=1000\). The core is heated beyond the Curie temperature \(T_{e}\). INCERT Exemplar] (a) The H field in the solenoid is (nearly) unchanged but the B field decreases drastically (b) The \(\mathrm{H}\) and \(\mathrm{B}\) fields in the solenoid are nearly unchanged (c) The magnetisation in the core reverses direction (d) The magnetisation in the core diminishes by a factor of ahout \(10^{1}\)

In an experiment with vibration magnetometer, the value of \(4 \pi^{2} I / T^{2}\) for a short bar magnet is observed as \(36 \times 10^{-4}\). In the experiment with deflection magnetometer with the same magnet, the value of \(4 \pi d^{3} / 2 \mu_{0}\) is observed as \(10^{8} / 36\). The magnetic moment of the magnet used is (a) \(50 \mathrm{~A}-\mathrm{m}\) (b) \(100 \mathrm{~A}-\mathrm{m}\) (c) \(200 \mathrm{~A}-\mathrm{m}\) (d) \(1000 \mathrm{~A}-\mathrm{m}\)

A vibration magnetometer consists of two identical bar magnets placed one over the other such that they are perpendicular and bisect each other. The time period of oscillation in a horizontal magnetic field is \(2^{5 / 4}\) s. One of the magnets is removed and if the other magnet oscillates in the same field, then the time period in second is (a) \(2^{1 / 4}\) (b) \(2^{10}\) (c) 2 (d) 4

Assertion-Reason type. Each of these contains two Statements: Statement 1 (Assertion), Statement II (Reason). Each of these questions also has four alternative choice, only one of which is correct. You have to select the correct choices from the codes (a), (b), (c) and (d) given below (a) If both Assertion and Reason are true and the Reason is correct explanation of the Assertion. (b) If both Assertion and Reason are true but Reason is not correct explanation of the Assertion. (c) If Assertion is true but Reason is false. (d) If Assertion is false but the Reason is true. Assertion Magnetic dipole possesses maximum potential energy when magnetie moment and magnetic field are parallel to each other. Reason Current loop is treated as a magnetic dipole.
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