Angular Momentum, Magnetic Dipole Moments MCQ Quiz - Objective Question with Answer for Angular Momentum, Magnetic Dipole Moments - Download Free PDF

The correct answer is - n

Key Points

• Magnetic moment
  • The magnetic moment due to the motion of an electron in an atom is directly related to its orbital angular momentum.
• Bohr's model
  • According to Bohr's model, the orbital angular momentum of an electron in the nth energy state is quantized and given by L=nℏ" id="MathJax-Element-1-Frame" role="presentation" style="position: relative;" tabindex="0">L=nℏL=nℏ $L = n\hbar$ , where ℏ" id="MathJax-Element-2-Frame" role="presentation" style="position: relative;" tabindex="0">ℏℏ $\hbar$ is the reduced Planck's constant.
• Proportionality
  • The magnetic moment is proportional to the orbital angular momentum, hence it is also proportional to n" id="MathJax-Element-3-Frame" role="presentation" style="position: relative;" tabindex="0">nn $n$ .

Additional Information

• Bohr's quantization condition
  • Bohr postulated that electrons move in orbits of fixed size and energy, and the angular momentum is quantized as L=nℏ" id="MathJax-Element-4-Frame" role="presentation" style="position: relative;" tabindex="0">L=nℏL=nℏ $L = n\hbar$ .
  • This quantization leads to discrete energy levels and magnetic moments associated with each energy state.
• Magnetic dipole moment
  • The magnetic dipole moment μ" id="MathJax-Element-5-Frame" role="presentation" style="position: relative;" tabindex="0">μμ $\mu$ due to an electron's orbital motion is given by μ=e2mL" id="MathJax-Element-6-Frame" role="presentation" style="position: relative;" tabindex="0">μ=e2mLμ=e2mL $\mu = \frac{e}{2m} L$ , where e" id="MathJax-Element-7-Frame" role="presentation" style="position: relative;" tabindex="0">ee $e$ is the electron charge, and m" id="MathJax-Element-8-Frame" role="presentation" style="position: relative;" tabindex="0">mm $m$ is its mass.
  • Since L=nℏ" id="MathJax-Element-9-Frame" role="presentation" style="position: relative;" tabindex="0">L=nℏL=nℏ $L = n\hbar$ , the magnetic moment μ" id="MathJax-Element-10-Frame" role="presentation" style="position: relative;" tabindex="0">μμ $\mu$ is directly proportional to n" id="MathJax-Element-11-Frame" role="presentation" style="position: relative;" tabindex="0">nn $n$ .
• Energy levels in hydrogen atom
  • In a hydrogen atom, the energy of an electron in the nth orbit is given by En=−13.6 eVn2" id="MathJax-Element-12-Frame" role="presentation" style="position: relative;" tabindex="0">En=−13.6 eVn2En=−13.6 eVn2 $E_n = -\frac{13.6 \text{ eV}}{n^2}$ .
  • Despite the n2" id="MathJax-Element-13-Frame" role="presentation" style="position: relative;" tabindex="0">n2n2 $n^2$ dependence of energy, the magnetic moment due to orbital motion depends linearly on n" id="MathJax-Element-14-Frame" role="presentation" style="position: relative;" tabindex="0">nn $n$ .

Calculation:

σ is constant in y-direction

So, y_cm = b/2

\(\mathrm{x}_{\mathrm{cm}}=\frac{\int_{0}^{\mathrm{a}} \mathrm{xdm}}{\int_{0}^{\mathrm{a}} \mathrm{dm}}\)

= \(\frac{\int_{0}^{a} \mathrm{x} \sigma_{\mathrm{x}} \mathrm{dA}}{\int_{0}^{\mathrm{a}} \sigma_{\mathrm{x}} \mathrm{dA}}\)

= \(\frac{\int_{0}^{a} x \frac{\sigma_{0} x}{a b} b d x}{\int_{0}^{a} \frac{\sigma_{0} x}{a b} b d x}\) 

\(\mathrm{x}_{\mathrm{cm}}=\frac{\int_{0}^{\mathrm{a}} \mathrm{x}^{2} \mathrm{dx}}{\int_{0}^{\mathrm{a}} \mathrm{xdx}}\)

= \(\frac{\left(\frac{x^{3}}{3}\right)_{0}^{a}}{\left(\frac{x^{2}}{2}\right)_{0}^{a}}=\frac{a^{3} / 3}{a^{2} / 2}\)

= \(\frac{2 \mathrm{a}}{3}\)

Concept Used:

The dipole moment (p) is given by the product of the magnitude of charge (Q) and the separation distance (d).

For a square, the relation between the diagonal (d) and side length (s) is:

s = d / √2

Calculation:

Given:

Mass of the aluminium coin, m = 0.75 g

Diagonal of the square coin, d = 14 mm = 14 × 10-3 m

Charge on the dipole, Q = +Q and -Q

Side of the square:

⇒ s = (14 × 10^-3) / √2

⇒ s = (14 / 1.414) × 10^-3

⇒ s = 9.9 × 10^-3 m

Dipole moment:

⇒ p = Q × s

Given that the charge Q is unknown, the dipole moment will be expressed in terms of Q:

⇒ p = Q × 9.9 × 10^-3 m

Converting into cm:

⇒ p = Q × 9.9 cm

Approximating the nearest correct answer choice:

∴ The correct answer is 348 cm.

Now angular displacement \(\theta= 0 +\dfrac{\alpha t^2}{2}=96\ rad\)

The magnetic moment is given by:

\(\mu=g\frac{q}{2m}S\)

where g is the gyromagnetic ratio,

q is the charge,

m is the mass, 

S is the spin.

All the three particles electrons, proton, and neutrons have \(\frac{1}{2}\) spin. 

The magnetic moment is ∝ \(\frac{1}{m}\).

Thus smaller the mass maximum the is the magnitude of the magnetic moment.

Thus \(\mu_e>\mu_P>\mu_n\) .

The correct option is (1).

The correct answer is two.

CONCEPT:

• Spin quantum number: The spin quantum number describes the angular momentum and direction of the spin of an electron in the orbital.
  • ​Each orbital can will hold two electrons. The two electrons will have +1/2 and -1/2 spin.
  • The electrons tend to fill the orbitals before they pair up.
  • Hence, the first electron filled in the orbital will have a spin of +1/2. The electrons start pairing up once all the orbitals are half-filled by one electron each.
  • The second electron to fill the orbital will have a spin of -1/2.

​EXPLANATION:

• The spin quantum number is used to determine the orientation of the spin of the electron i.e. either a spin up or spin down.
• This orientation is referred to with +1/2 for a spin up and -1/2 for spin down.
• Hence, an electron spin quantum number can have only two values.

CONCEPT:

• When a circular loop is associated with the current I, it starts to act as a magnet and its magnetic moment is find as given below.

• Magnetic moment (μ): The magnetic strength and orientation of a magnet or other object that produces a magnetic field.
  • It is a vector quantity associated with the magnetic properties of electric current loops.
  • It is equal to the amount of current flowing through the loop multiplied by the area encompassed by the loop.

μ = N i A 

where μ is the magnetic moment, A is the area of the coil, N is no. of turns and I is current in the coil.

• Its direction is established by the right-hand rule for rotations.

EXPLANATION:

• From the above, it is clear that a circular loop is considered as a magnetic dipole, then the dipole moment is the product of current and area.

Concept:

• If the circular loop is considered as a magnetic dipole, then the dipole moment is the product of current and area. But the circular loop has multiple numbers of turns.
• Therefore, the magnitude of dipole moment = area × current × number of turns.

i.e. m = NIA = NI (π r²)

where, Magnetic moment of copper coil = m

No of loops = N

Current flowing though loop = I

Area of coil = r

Explanation:

From the above explanation we can see that, magnetic moment of circular coil with N number of terms is expressed as 

m = NIA

Now for a sing loop the above equation can be expressed as 

m = IA 

Concept:

Diamagnetic materials 

• Diamagnetic materials are those in which the individual atoms do not possess any permanent magnetic dipole moments.
• The atoms of such material however acquire an induced dipole moment when they are placed in an external magnetic field. 
• Diamagnetic materials repeal the magnetic field weakly, results from the orbital motion of electrons. 

• Each circulating electrons act as a current loop which produces a magnetic field.
• Two electrons travel in each orbit and are in opposite directions.
• The magnetic moment produced by each electron of the orbit will cancel. that's why diamagnetic materials have no residual magnetization. 

Explanation:

• The presence of an external magnetic field affects the orbital motion of the electrons in an atom in such a  way that the atom generates a magnetic field that opposes the external field. This is referred to as diamagnetism.

So, diamagnetism arises due to the orbital motion of electrons. 

CONCEPT: 

• Moving Charge produces Magnetic Field.
• Magnetic Moment of an Atom (μ): Magnetic Moment of an Atom is a measure of the strength of the magnetic field produced by the orbital angular momentum of an electron. 

Magnetic Moment (μ) of an Electron is given as

μ = (πeν²r)/ 2

Where e = electronic charge = 1.60217662 × 10^-19 coulombs, ν = Frequency of Electron by which it is rotating Around Nucleus, and r = radius of atom

CALCULATION:

Given,

Frequency ν =  6.8 × 1015 c/s

radius of atom r =  5.1 × 10-11 m

So, \(μ = \frac{(1.602 \times 10^{-19})\times \pi \times (6.8\times10^{15})( 5.1\times10^{-11})}{2}^2\)

Magnetic moment (μ) = 8.9 × 10-24 A x m2

CONCEPT:​

• Angular momentum: the quantity of a rotating body, which is the product of its moment of inertia and its angular velocity.

Angular momentum (L) = m v r

where m is the mass of the body, v is velocity and r is the distance from the rotating point.

• Bohr proposed that the angular momentum L of an electron in its orbit is quantized and it has only specific discrete values.

From Bohr's Model of Atom, n^th orbit has angular momentum:

​angular momentum (L) = mvr = \( {nh \over 2π}\)

where n is the orbit number, h is plank constant.

CALCULATION:

• From Bohr's Model of Atom, nth orbit has angular momentum

​angular momentum (L) = \( {nh \over 2π}\)

• So the correct answer is option 3.

The correct answer is two.

CONCEPT:

• Spin quantum number: The spin quantum number describes the angular momentum and direction of the spin of an electron in the orbital.
  • ​Each orbital can will hold two electrons. The two electrons will have +1/2 and -1/2 spin.
  • The electrons tend to fill the orbitals before they pair up.
  • Hence, the first electron filled in the orbital will have a spin of +1/2. The electrons start pairing up once all the orbitals are half-filled by one electron each.
  • The second electron to fill the orbital will have a spin of -1/2.

​EXPLANATION:

• The spin quantum number is used to determine the orientation of the spin of the electron i.e. either a spin up or spin down.
• This orientation is referred to with +1/2 for a spin up and -1/2 for spin down.
• Hence, an electron spin quantum number can have only two values.

CONCEPT:

• When a circular loop is associated with the current I, it starts to act as a magnet and its magnetic moment is find as given below.

• Magnetic moment (μ): The magnetic strength and orientation of a magnet or other object that produces a magnetic field.
  • It is a vector quantity associated with the magnetic properties of electric current loops.
  • It is equal to the amount of current flowing through the loop multiplied by the area encompassed by the loop.

μ = N i A 

where μ is the magnetic moment, A is the area of the coil, N is no. of turns and I is current in the coil.

• Its direction is established by the right-hand rule for rotations.

EXPLANATION:

• From the above, it is clear that a circular loop is considered as a magnetic dipole, then the dipole moment is the product of current and area.

Concept:

• If the circular loop is considered as a magnetic dipole, then the dipole moment is the product of current and area. But the circular loop has multiple numbers of turns.
• Therefore, the magnitude of dipole moment = area × current × number of turns.

i.e. m = NIA = NI (π r²)

where, Magnetic moment of copper coil = m

No of loops = N

Current flowing though loop = I

Area of coil = r

Explanation:

From the above explanation we can see that, magnetic moment of circular coil with N number of terms is expressed as 

m = NIA

Now for a sing loop the above equation can be expressed as 

m = IA 

Concept:

Diamagnetic materials 

• Diamagnetic materials are those in which the individual atoms do not possess any permanent magnetic dipole moments.
• The atoms of such material however acquire an induced dipole moment when they are placed in an external magnetic field. 
• Diamagnetic materials repeal the magnetic field weakly, results from the orbital motion of electrons. 

• Each circulating electrons act as a current loop which produces a magnetic field.
• Two electrons travel in each orbit and are in opposite directions.
• The magnetic moment produced by each electron of the orbit will cancel. that's why diamagnetic materials have no residual magnetization. 

Explanation:

• The presence of an external magnetic field affects the orbital motion of the electrons in an atom in such a  way that the atom generates a magnetic field that opposes the external field. This is referred to as diamagnetism.

So, diamagnetism arises due to the orbital motion of electrons.