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The magnetic field at a point at a distance 'd' from the centre on axial line of a very short bar magnet of magnetic moment M is B. The magnetic field at a distance '2d' from centre on equitorial line of magnet of magnetic moment 8 M, will be
  • a)
    2 B
  • b)
    4 B
  • c)
    B/2
  • d)
    B/4
Correct answer is option 'C'. Can you explain this answer?
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The magnetic field at a point at a distance d from the centre on axial...
Given:
- A very short bar magnet of magnetic moment M
- Magnetic field at a point at a distance d from the centre on axial line of the magnet is B
- Magnetic field at a distance 2d from centre on equatorial line of magnet of magnetic moment 8 M is to be found.

To find: The magnetic field at a distance 2d from centre on equatorial line of magnet of magnetic moment 8 M.

Solution:
Let's consider the magnetic field at a point P on the equatorial line of the magnet at a distance 2d from the centre.

Magnetic field at point P due to the short bar magnet of magnetic moment M:
- Consider a small element dl of the bar magnet at a distance x from the centre on the equatorial line.
- Magnetic field at point P due to this small element dl is given by the formula:
dB = μ₀/4π * (2M/x³) * dl sinθ
where μ₀ is the permeability of free space, θ is the angle between the line joining dl and P and the axial line of the magnet.

- As we move along the equatorial line, the angle θ changes from 0 to 90 degrees.
- At the centre of the magnet (x = 0), the magnetic field due to all the small elements on the magnet cancel out.
- At a distance 2d from the centre (x = 2d), the angle θ is 90 degrees and sinθ = 1.
- Hence, the magnetic field at point P due to the short bar magnet of magnetic moment M is given by:
B₁ = μ₀/4π * (2M/(2d)³) * ∫dl = μ₀/4π * M/d³ * ∫dl

- The integral ∫dl represents the total magnetic moment of the bar magnet which is equal to M.

- Therefore, B₁ = μ₀/4π * M/d³ * M = μ₀/4π * (M/d³)^2

Magnetic field at point P due to the short bar magnet of magnetic moment 8M:
- Since the magnetic moment is 8 times larger, the magnetic field at point P due to this magnet will be 8 times larger than B₁.
- Hence, the magnetic field at a distance 2d from centre on equatorial line of magnet of magnetic moment 8 M is:
B₂ = 8B₁ = 8 * μ₀/4π * (M/d³)^2
B₂/B = 8 * μ₀/4π * (M/d³)^2 / μ₀/4π * (M/d²) = (M/d)² / (M/d³)² = 1/4

Therefore, the magnetic field at a distance 2d from centre on equatorial line of magnet of magnetic moment 8 M is B/4 or B/2 in absolute value (since the direction of the magnetic field is opposite to that of B). Hence, the correct option is (c) B/2.
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The orbital and spin angular momentum of the atom influence its magnetic structure and these properties are most directly studied by placing the atom in a magnetic field. Also, a magnetic field can affect the wavelengths of the emitted photons.The angular momentum vector associated with an atomic state can take up only certain specified directions in space. This concept of space quantization was shown by Otto Stern and Walthor Gerlach in their experiment.In the experiment, silver is vapourized in an electric oven and silver atoms spray into the evacuated apparatus through a small hole in the oven wall. The atoms which are electrically neutral but have a magnetic moment, are formed into a narrow beam as they pass through a slit in a screen. The beam, thus collimated, then passes between the poles of an electromagnet and finally, deposits its silver atoms on a glass plate that serves as a detector. The pole faces of the magnet are shaped to make the magnetic field as nonuniform as possible.In a non-uniform magnetic field, there is a net force on a magnetic dipole. Its magnitude and direction depends on the orientation of the dipole. Thus the silver atoms in the beam are deflected up or down, depending on the orientation of their magnetic dipole moments with respect to the z–direction.The potential energy of a magnetic dipole in a magnetic field where is magnetic dipole moment of the atom. From symmetry, the magnetic field at the beam position has no x or y components i.e.The net force Fz on the dipole isThus, the net force depends, not on the magnitude of the field itself, but on its spatial derivative or gradient.The ResultsIf space quantization did not exist, then could take on any value from + to –, the result would be a spreading out of the beam when the magnet was turned ON. However, the beam was split cleanly into two subbeams, each subbeam corresponding to one of the two permitted orientations of the magnetic moment ofthe silver atom, as shown.In a silver atom, all the spin and orbital magnetic moments of the electrons cancel, except for those of the atoms single valance electron. For this electron the orbital magnetic moment is zero because orbital angular momentum is zero (because for electrons of s–orbit, L = 0), leaving only the spin magnetic moment. This can take up only two orientations in a magnetic field, corresponding to ms = +1/2 and ms = – 1/2. Hence there are two subbeams – and not some other number.Q.A hydrogen atom in ground state passes through a magnetic field that has a gradient of 16mT/m in the vertical direction. If vertical component magnetic moment of the atom is 9.3 × 10–24 J/T, then force on it due to the magnetic moment of the electron is

The orbital and spin angular momentum of the atom influence its magnetic structure and these properties are most directly studied by placing the atom in a magnetic field. Also, a magnetic field can affect the wavelengths of the emitted photons.The angular momentum vector associated with an atomic state can take up only certain specified directions in space. This concept of space quantization was shown by Otto Stern and Walthor Gerlach in their experiment.In the experiment, silver is vapourized in an electric oven and silver atoms spray into the evacuated apparatus through a small hole in the oven wall. The atoms which are electrically neutral but have a magnetic moment, are formed into a narrow beam as they pass through a slit in a screen. The beam, thus collimated, then passes between the poles of an electromagnet and finally, deposits its silver atoms on a glass plate that serves as a detector. The pole faces of the magnet are shaped to make the magnetic field as nonuniform as possible.In a non-uniform magnetic field, there is a net force on a magnetic dipole. Its magnitude and direction depends on the orientation of the dipole. Thus the silver atoms in the beam are deflected up or down, depending on the orientation of their magnetic dipole moments with respect to the z–direction.The potential energy of a magnetic dipole in a magnetic field where is magnetic dipole moment of the atom. From symmetry, the magnetic field at the beam position has no x or y components i.e.The net force Fz on the dipole isThus, the net force depends, not on the magnitude of the field itself, but on its spatial derivative or gradient.The ResultsIf space quantization did not exist, then could take on any value from + to –, the result would be a spreading out of the beam when the magnet was turned ON. However, the beam was split cleanly into two subbeams, each subbeam corresponding to one of the two permitted orientations of the magnetic moment ofthe silver atom, as shown.In a silver atom, all the spin and orbital magnetic moments of the electrons cancel, except for those of the atoms single valance electron. For this electron the orbital magnetic moment is zero because orbital angular momentum is zero (because for electrons of s–orbit, L = 0), leaving only the spin magnetic moment. This can take up only two orientations in a magnetic field, corresponding to ms = +1/2 and ms = – 1/2. Hence there are two subbeams – and not some other number.Q.A hydrogen atom in ground state passes through a magnetic field that has a gradient of 16mT/m in the vertical direction. If vertical component magnetic moment of the atom is 9.3 × 10–24 J/T, then force on it due to the magnetic moment of the electron is

The orbital and spin angular momentum of the atom influence its magnetic structure and these properties are most directly studied by placing the atom in a magnetic field. Also, a magnetic field can affect the wavelengths of the emitted photons.The angular momentum vector associated with an atomic state can take up only certain specified directions in space. This concept of space quantization was shown by Otto Stern and Walthor Gerlach in their experiment.In the experiment, silver is vapourized in an electric oven and silver atoms spray into the evacuated apparatus through a small hole in the oven wall. The atoms which are electrically neutral but have a magnetic moment, are formed into a narrow beam as they pass through a slit in a screen. The beam, thus collimated, then passes between the poles of an electromagnet and finally, deposits its silver atoms on a glass plate that serves as a detector. The pole faces of the magnet are shaped to make the magnetic field as nonuniform as possible.In a non-uniform magnetic field, there is a net force on a magnetic dipole. Its magnitude and direction depends on the orientation of the dipole. Thus the silver atoms in the beam are deflected up or down, depending on the orientation of their magnetic dipole moments with respect to the z–direction.The potential energy of a magnetic dipole in a magnetic field where is magnetic dipole moment of the atom. From symmetry, the magnetic field at the beam position has no x or y components i.e.The net force Fz on the dipole isThus, the net force depends, not on the magnitude of the field itself, but on its spatial derivative or gradient.The ResultsIf space quantization did not exist, then could take on any value from + to –, the result would be a spreading out of the beam when the magnet was turned ON. However, the beam was split cleanly into two subbeams, each subbeam corresponding to one of the two permitted orientations of the magnetic moment ofthe silver atom, as shown.In a silver atom, all the spin and orbital magnetic moments of the electrons cancel, except for those of the atoms single valance electron. For this electron the orbital magnetic moment is zero because orbital angular momentum is zero (because for electrons of s–orbit, L = 0), leaving only the spin magnetic moment. This can take up only two orientations in a magnetic field, corresponding to ms = +1/2 and ms = – 1/2. Hence there are two subbeams – and not some other number.Q.A hydrogen atom in ground state passes through a magnetic field that has a gradient of 16mT/m in the vertical direction. If vertical component magnetic moment of the atom is 9.3 × 10–24 J/T, then force on it due to the magnetic moment of the electron is

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The magnetic field at a point at a distance d from the centre on axial line of a very short bar magnet of magnetic moment M is B. The magnetic field at a distance 2d from centre on equitorial line of magnet of magnetic moment 8 M, will bea)2 Bb)4 Bc)B/2d)B/4Correct answer is option 'C'. Can you explain this answer?
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The magnetic field at a point at a distance d from the centre on axial line of a very short bar magnet of magnetic moment M is B. The magnetic field at a distance 2d from centre on equitorial line of magnet of magnetic moment 8 M, will bea)2 Bb)4 Bc)B/2d)B/4Correct answer is option 'C'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about The magnetic field at a point at a distance d from the centre on axial line of a very short bar magnet of magnetic moment M is B. The magnetic field at a distance 2d from centre on equitorial line of magnet of magnetic moment 8 M, will bea)2 Bb)4 Bc)B/2d)B/4Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The magnetic field at a point at a distance d from the centre on axial line of a very short bar magnet of magnetic moment M is B. The magnetic field at a distance 2d from centre on equitorial line of magnet of magnetic moment 8 M, will bea)2 Bb)4 Bc)B/2d)B/4Correct answer is option 'C'. Can you explain this answer?.
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