Mathematics Definition Based Problems

### Definition & Proof Based Based Problems

Q 3232423332

What is figure of merit of a galvanometer?

Solution:

Figure of merit is defined as the amount of current which produces unit deflection in the galvanometer.
Q 3262234135

Define the current sensitivity of a moving coil galvanometer. "Increasing the current sensitivity may not necessarily increase the voltage sensitivity." Justify this statement.

Solution:

Current Sensitivity: It is defined as the deflection produced in a coil per unit current passed through it.

I_s = alpha/I = (NBA)/k

I = (BAN)/K ; if N -> 2N , the l -> 2 l and R -> 2 R

 :. V_s = (BAN)/(K R) = (BA (2N))/(K (2R)) = (BAN)/(K R)

i.e. if the current sensitivity is doubled, say by doubling the number of turns, then the voltage sensitivity may not be increased because it will increase the resistance of the galvanometer and the voltage sensitivity may remain the same.
Q 3222012831

Define one tesla using the expression for the magnedc force acting on a particle of charge q moving with velocity vec v in a magnetic field vec B.

Solution:

As F_text(max) = qvB => B = F_text(max) /(qv)

If F_text( max) = 1 N, q = + 1 C and v = 1 m//s and theta = 90°, then B = 1 T

Hence, one tesla is the magnetic field in which a normally entering + 1 C

charge, moving at 1 ms^(-1) experiences a maximum force of 1 N.
Q 3242423333

Define gyromagnetic ratio.

Solution:

It is the ratio of the magnetic dipole moment to the angular momentum of the electron revolving round the nucleus.
Q 3262234135

Define the current sensitivity of a moving coil galvanometer. "Increasing the current sensitivity may not necessarily increase the voltage sensitivity." Justify this statement.

Solution:

Current Sensitivity: It is defined as the deflection produced in a coil per unit current passed through it.

I_s = alpha/I = (NBA)/k

I = (BAN)/K ; if N -> 2N , the l -> 2 l and R -> 2 R

 :. V_s = (BAN)/(K R) = (BA (2N))/(K (2R)) = (BAN)/(K R)

i.e. if the current sensitivity is doubled, say by doubling the number of turns, then the voltage sensitivity may not be increased because it will increase the resistance of the galvanometer and the voltage sensitivity may remain the same.
Q 3272234136

Deduce the expression for the magnetic dipole moment of an electron orbiting around the central nucleus.

Solution:

Consider an electron revolving around the nucleus of an atom. The electron is in a uniform circular motion around the nucleus of charge +Ze. This constitutes a current.

:. I = e/T  ............(i)

If r is the orbital radius of the electron and 'v' is the orbital speed, then the time period is given as

 T = (2 pi r)/v .............(ii)

From equations (i) and (ii), we get

 I = (ev)/(2 pi r) .............(iii)

As the magnetic moment is given by
mu_l = I pi r^2

We have  mu_l = ( (ev)/(2 pi r)) pi r^2 = (evr)/2

Multiplying and dividing by me in above equation, we get

mu_l = (evm_e r)/( 2 m_e) = e/(2 m_e) l

where l is the angular momentum of the electron. According to the Bohr hypothesis, the angular momentum can have discrete values only.

i.e  l = (nh)/(2 pi)

So,  mu_l = (enh)/(2 pi ( 2 m)) = ( n e h)/(4 pi m)

In this case it is directed into the plane of the paper.
Q 3212823730

Write the expression for Lorentz magnetic force on a particle of charge q moving with velocity vec v in a magnetic field vec B. Show that no work is done by this force on the charged particle.

Solution:

Lorentz magnetic force, vec F = q ( vec v xx vec B)

 ∵ Work done = vec F . vec S. (∵ vec S || vec v)

As the force is acting in a direction normal to the velocity of a charged particle (i.e. displacement), no work is done by the force.
Q 3212012830

Write the expression, in a vector form, for the Lorentz magnetic force vec F due to a charge moving with velocity vec v in a magnetic field vec B. What is the direction of the magnetic force ?

Solution:

Lorentz magnetic force ( vec F_m ) = q ( vec v xx vec B). The direction of magnetic force is perpendicular to the plane containing velocity and magnetic field vectors.
Q 3232423332

What is figure of merit of a galvanometer?

Solution:

Figure of merit is defined as the amount of current which produces unit deflection in the galvanometer.
Q 3242823733

What is velocity selector? Write its uses.

Solution:

Velocity selector is a device consisting of perpendicular electric and magnetic fields that can be used as a velocity filter for charged particles. It is used to measure charge to mass ratio and is also used in Mass Spectrometer.
Q 3282223137

Write the expression for the magnetic moment of a circular coil of area A carrying a current I, in a vector form.

Solution:

vec M = I vec A, as vector vec A is perpendicular to the surface, the magnetic moment vec M will also be perpendicular to the plane of circular coil.