MOTIONAL emf

The magnetic flux linked with a coil can be changed in many ways. For this,
( 1 ) the magnet can be moved with respect to the coil.
( 2 ) the coil can be rotated in the magnetic field.
( 3 ) the coil can be kept inside the magnetic field in proper manner and the magnitude of the magnetic induction can be changed.
( 4 ) the coil can be moved inside a non-uniform magnetic field.
( 5 ) the dimensions of the coil placed inside a magnetic field can somehow be changed.
“The induced emf. produced due to the change in magnetic flux linked with a coil due to some kind of motion is called motional emf.”
An example of motional emf. Is illustrated as under.
A U-shaped conducting wire is placed in a
plane perpendicular to the magnetic field. The magnetic field lines enter the plane of paper as shown by ( + ) sign.
A conducting rod is slid over the two arms of
the conductor with a constant velocity v. The
perpendicular distance between the two arms
of the conductor is l.
MN is the position of the rod at time t when the magnetic flux linked with the loop PMNO is
Flux = = ( area of PMNO ) ´ B ( magnetic field intensity )
= l B x, where x = PM = ON
As the emf. Is generated due to the motion of the rod, it is known as motional emf. Motional emf. Is produced by a conducting rod moving in a magnetic field in appropriate manner even without the  U-shaped conductor. This is explained in the following example with the reason behind the origin of induced emf.
The reason behind the origin of induced emf
With the conducting rod moving with velocity v , positive ions and electrons in it also move perpendicularly to the magnetic field B . Electrons move from Q to P under the effect of Lorentz force
F = q v * B leaving positive ions exposed at Q. Thus rod behaves as a battery of emf B v l.
Conversion of Mechanical Energy into Electrical Energy
In the above example, let I  be the current flowing through the sliding conductor. As it is moving in a magnetic field entering the plane of paper, conventional current flows in the rod from P to Q and the rod experiences force I B l in the direction opposite to its velocity v. Thus to maintain its uniform velocity, a force of magnitude I B l must be applied in the direction of its velocity. Such a force is called Lenz force.
Hence, mechanical power, Pm = force ´ velocity = Fv = B I l v
and electrical power, Pe = voltage ´ current = B v l * I   = B I l v
Thus, mechanical power spent is converted into electrical power. Here ideal case of zero circuit resistance is considered.

FARADAY'S LAW OF ELECTROMAGNETIC INDUCTION

“The induced emf. produced in a closed circuit ( or a coil ) is equal to the negative of the rate
of change of magnetic flux linked with it.”
Thus, average induced emf. produced, < e > = - N( Flux) / time
and the instantaneous induced emf. at time t, e  = - lim N (Flux) / time = - N ( d Flux / d time)
that is the derivative of Flux with respect of time
where, N = number of turns of the coil,

LENZ'S LAW OF ELECTROMAGNETIC INDUCTION

As shown in the figure, suppose a bar magnet is moved towards a conducting coil with its north pole facing the coil. If this produces current in the coil in the clockwise direction as seen from the side of the magnet, then the side of the coil facing the magnet will act like south pole of a magnet and will attract the magnet. The magnet will have accelerated motion towards the coil which will increase the rate of change of flux and hence the current in the coil. This will increase the force of attraction and the acceleration of the magnet will increase further. Thus the current in the coil will go on increasing. If a resistance R is connected in the coil, joule heat I2Rt is produced in it. No mechanical work is done in giving a slight push to the magnet. Thus heat energy is being continuously produced without spending energy. This is contrary to the law of conservation of energy. Thus our assumption about the direction of current induced in the coil being clockwise is incorrect.
If the direction of induced current were counter-clockwise, the end of the coil facing the north pole of the magnet would have become north pole and mechanical work will be required to be done against the force of repulsion which gets converted in the joule heat in the resistance of the coil. This is consistent with the law of conservation of energy.
Thus “induced emf ( or induced current ) is produced in such a direction that the magnetic field produced due to it opposes the very cause ( here motion of the magnet ) that produces it”. This statement is known as Lenz’s law.

FARADAY'S EXPERIMENTS OF ELECTROMAGNETIC INDUCTION

Faraday took a ring of soft iron. On one side of it, an insulated conducting coil was connected with a battery. On the opposite side, another conducting
coil was connected with a galvanometer.
Faraday observed that passing a steady current through the left coil produced no effect on the galvanometer in the right coil. However, a momentary deflection of galvanometer was noticed whenever the battery was switched on or off. When a steady current is passed magnetic flux produced in the left coil passes through the right coil which does not produce any current in it. Whenever the battery is switched on or off, magnetic flux in the right coil changes from zero to maximum or maximum to zero
respectively. This rate of change of magnetic flux in the right coil produces current in it. In another experiment, Faraday arranged two bar magnets in the shape of V. At the open end of V, he kept one soft iron rod with an insulated copper wire wound around it to which galvanometer was connected. On moving the upper magnet up and down, galvanometer showed deflection. Magnetic flux through the coil increased when the magnet touched  the iron rod and decreased when it moved away. Faraday concluded from these experiments that ‘To produce electric field in a coil, the change in magnetic flux is important and not the flux itself.’
Faraday also noted that:
( i ) More current is produced when the magnet is moved faster due to faster change of
magnetic flux linked with the coil.
( ii ) When a coil carrying electric current is placed  above another coil and relative motion produced between the two coils, galvanometer shows  deflection in the other coil.
( iii ) If any of the two coils is rotated with respect to the other, then also galvanometer shows deflection.
( iv ) If the north pole of a bar magnet is
moved towards a coil, the galvanometer
shows deflection. Now if the magnet is
moved away from the coil, the galvanometer
shows deflection in the opposite direction.
Similar results are obtained with the south pole of  the magnet with deflections of galvanometer in  opposite direction.
Faraday named the current produced as the ‘induced current’, the emf as ‘induced emf’ and the phenomenon as ‘magnetic induction’.

HEATING EFFECT OF ELECTRIC CURRENT: JOULE'S LAW

The electric current in a conductor is due to the motion of electrons. During their motion, electrons collide with the oscillating positive ions in the conductor and impart part of their energy to them. Ions oscillate faster and their increased energy is manifested as heat. The heat energy released in a conductor on passing an electric current is called the “Joule heat” and effect is called the ‘Joule effect”. The potential difference of V volt applied between two ends of a conductor means that V joule of electrical energy is utilized and converted into heat when one coulomb charge passes through the conductor. If Q coulomb charge passes through the conductor in t seconds resulting in current I, the heat energy produced is
W = V Q 
    = V I t
    = I 2 R t         ( Q V = I R according to Ohm’s law )
    = ( V 2/ R ) 
The electric power, i. e., the electrical energy supplied per unit time or converted into heat energy per unit time in a resistance R, is
P = V I      
   = I 2 R
   = ( V 2/ R
Thus, mechanical unit of energy, joule = watt. second which is an electrical unit of energy. This being too small, kilowatt-hour ( kwh ) = 3.6 × 106 joule is used as a practical unit of electrical energy. R is the Ohmic resistance of the conductor value of which does not depend upon V or I. Considering R as a constant,
P is proportional to  I 2 or P is proportional to V 2. In fact, all electrical appliances are rated to operate for a given potential difference and hence in household wiring, they are connected in parallel. Therefore, V is same for all whereas I varies.
Joule’s Law:  - “The heat produced per unit time, on passing electric current through a conductor at a given temperature, is directly proportional to the square of the electric current”.
To express heat produced in calories, the following relation given by Joule is used.
W = JH  
where W is mechanical energy in joule,
H is heat energy in calorie
and J = 4.2 joule / calorie is Joule’s constant or mechanical equivalent of heat. 

CAPACITORS AND CAPACITANCE

When positive electric charge on an isolated conducting sphere shown in figure 1 is gradually increased, electric potential on its surface and electric field in its vicinity increase. When the electric field becomes strong, it ionizes the surrounding air which causes charge on the sphere to leak and the charge on the sphere cannot be increased further. During this process, the ratio of charge ( Q ) and the electric potential ( V ) of the sphere remains constant. This ratio, Q / V, is called its capacitance ( C ). To increase capacitance C of the sphere, another isolated conducting sphere is brought near it on which charge is induced as shown in figure. 2. On earthing, the positive charge gets neutralized as shown in figure 3. The negative charge induced on the second sphere reduces the electric potential of the first sphere thereby increasing its charge storage capacity. The ratio Q / V of the charge on the first sphere and the potential difference between the two spheres is still constant and is called the capacitance C of the system. The value of C depends on the dimensions of the spheres, the distance between the two spheres and the medium between them.  The arrangement in which two good conductors of arbitrary shape and volume, are arranged close to one another, but separated from each other, is called a capacitor. The  conductors are known as plates of the capacitor. Positively charged conductor is called the positive plate and the negatively charged conductor the negative plate. Both the plates are equally charged. The charge  on the positive plate is called the charge ( Q ) of the capacitor and taking the potential difference between the two plates as V, capacitance of the capacitor is C = Q / V. The S.I. unit of capacitance is coulomb / volt which is also called farad ( F ) named after the great scientist Michael Faraday. The smaller units of farad are microfarad ( m F = 10 (exp- 6) F ) and picofarad ( pF = 10 (exp-12) F ).
Parallel Plate Capacitor: -

This type of capacitor is made by two metallic plates having identical area and kept parallel to each other. The distance ( d ) between the two plates is kept less as compared to the dimensions of the plates to minimize non-uniform electric field due to the irregular distribution of charges near the edges.
Let Q = electric charge on the capacitor
∴σ  = Q / A = surface charge density
As d is very small, the plates can be considered as infinitely charged planes and the electric field between the plates can therefore be considered uniform.The electric fields, E1 and E2, between the plates due to positively charged and negatively charged plates respectively are equal in magnitude and direction. The direction of both E1 and E2 is from the positively charged plate to the negatively charged plate. The resultant electric field between the plates is, therefore, 

E = E1 + E2 =  σ    +    σ     = σ / ε0  = Q / ε0A (because  σ = Q / A)

                      2ε0      2ε0  

Outside the plates, E1 and E2 being oppositely directed cancel each other resulting in zero electric field in this region. The potential difference between the two plates is

V = E d = Q d / ε0 A

because the capacitance of the capacitor, C = Q /V = ε0 A / d

( a ) Series Connection of Capacitors: -

The end to end connection of capacitors as shown In the figure is called the series connection of capacitors. Equal charge Q deposits on each capacitor, but the p.d. between their plates is different depending on the value of its capacitance.
∴ V = V1 + V2 + …. + Vn

       = Q / C1 + Q / C2 + Q / C 3 + --------- + Q / C n

dividing both sides by Q we get the following equation i.e.

V / Q = 1 / C1 + 1 . C 2 + 1 / C3 + ------- + 1 / Cn When all capacitors connected in series are replaced by a single capacitor of capacitance C  such that the charge deposited on it is Q with the same voltage supply, then such a capacitor is called their equivalent capacitor. 

Now as V / Q = 1 / C (because Q / V = C)  so above equation becomes 

1 / C = 1 / C1 + 1 / C2 + 1/ C3 + ------- + 1 / Cn

The value of C is smaller than the smallest of C1, C2, …. Cn.

( b ) Parallel Connection of Capacitors: -

The connection of capacitors in which positive plates of all capacitors are connected to a single point and negative plates to another single point in a circuit is called parallel connection of capacitors as shown in the figure. In such a connection, charge accumulated on each of the capacitors is different depending on the value of its capacitance, but the p.d. across all is the same.

Thus, total charge Q = Q1 + Q2 + Q3 + ….
                                = ( C1 + C2 + C3 + …. ) V
When all capacitors connected in parallel are replaced by a single capacitor of capacitance C such that the charge deposited on it is Q with the same voltage supply, then such a capacitor is called their equivalent capacitor. Its value is
C = Q / V = C1 + C2 + C3 + ….