moving charges and magnetism

Who first noticed that a current in a straight wire caused a noticeable deflection in a nearby magnetic compass needle.

Hans Christian Oersted

moving charges or currents produced a

magnetic field in the surrounding space.

Radio waves were discovered by....and produced by....

Hertz
J.C.Bose and G. Marconi

Magnetic field superposition principle

the magnetic field of several sources is the vector addition of magnetic field of each individual source

Lorentz force

When a charge particle is under the influence of both electric and magnetic force than the combined effect of both is called Lorentz force
F=qE + qvBsinθ => F = q( E+ vBsinθ)

Magnetic force; 3 points on what magnetic force depends

F= q2v2Bsinθ
B= external magnetic field
1)It depends on q, v and B
2)The magnetic force q [ v × B ] includes a vector product of velocity and magnetic field. The vector product makes the force due to magnetic field vanish (become zero) if velocity and magnetic field are parallel or anti-parallel
3)The magnetic force is zero if charge is not
moving

Magnetic force

F = q2v2B sinθ
F= i2L2B sinθ
L= the length of the rod

1 Tesla

The magnitude of magnetic field B is 1 SI unit, when the force acting on a unit charge (1 C), moving perpendicular to B with a speed 1m/s, is one newton.

1 Gauss

10^-4 Tesla

Do magnetic force do work on moving charge particle (if it enters magnetic field at 90°)

No work is done ; because magnetic force and displacement are perpendicular to each other(but the direction of velocity change not magnitude)
Even when charge particle move parallel to magnetic field , than magnetic force is zero, hence W=0
So in helical path also W=0

Electric field or magnetic field, who can transfer energy in addition to momentum.]

electric field

When moving charged particle move in circular path under the influence of magnetic field

When it (charged particle) enters at 90° to the magnetic field (external)

Condition for moving in circular path

Force must be perpendicular to velocity

v (of charge particle)perpendicular to (external magnetic field)B. The perpendicular force, q v × B, acts as a) ....... and produces (b)....motion(c).......magnetic field

a) centripetal force
b) circular
c) perpendicular

Case when moving charged particle goes undeflected in external magnetic field

Velocity (of charge particle) is parallel or anti parallel to external magnetic field

When moving charge particle describe helical path

When moving charge particle enter at angle "θ" to the magnetic field

The larger the (a).....the larger is the radius and bigger the circle described.(of moving charge particle in magnetic field)

Momentum
r= mv/(qB)

Angular velocity (ω) [of moving charge particle in circular path in magnetic field] is independent of

Velocity/ energy
ω =2πν= qB/m

Angular velocity (of move circular path charge particle in magnetic field)

ω =2πν= qB/m

Time for 1 revolution (by charge particle moving in circular path under influence of magnetic field)

T= 2πm/ qB
B= external magnetic field

Radius of helicx

The radius of the circular component of motion

Principle of velocity filter/ selector

Cross field (when electric and magnetic field are perpendicular to each other)
E= vB
E= electric field
B= magnetic field
v= velocity of particle who undeflected
Only particles with speed E/B pass undeflected through the region of crossed fields.

Mass spectrometer

an instrument used to determine the relative masses of atoms by the deflection of their ions on a magnetic field
Principle- cross field
a device that separates charged particles, usually ions, according to their charge to mass ratio.

Important points related to cyclotron

1) principle- cross field, magnetic field change direction (not accelerate) and electric field accelerate the charged particle
2)Cyclotron uses the fact that the frequency of revolution of the charged particle in a magnetic field is independent of its energy
3) two semicircular disc-like metal containers, D1 and D2, which are called dees; inside D electric field is ZERO but magnetic field acts inside D's
4) with increase in energy, the radius of the circular path increases.
5)assembly is evacuated to minimise collisions between the ions and the air molecules. A high frequency alternating voltage is applied to the dees.
6)νa( applied voltage frequency)= νc h frequency of charge particle/ cyclotron)is called the resonance condition.
7)time for one revolution of an ion is independent of its speed or radius of its orbit.
8) used to study nuclear reactions, implant ions into solids and modify their properties or even synthesise new materials,hospitals to produce radioactive substances which can be used in diagnosis and treatment

Electric field can ......charge particle but magnetic field cannot
Magnetic field can change ....of charged particle but electric field cannot

Accelerate
Direction

Biot-Savart Law

dB = μ0 i dl sinθ/ (4π r2)

Relationship between permittivity, permeability, speed of light

ε0μ0= (1/c^2)

The Biot-Savart law for the magnetic field has certain similarities, as well as, differences with the Coulomb's law

1) Both are long range, since both depend inversely on the square; The principle of superposition applies to both fields
2)electrostatic field is produced by a scalar source; The magnetic field is produced by a vector source I dl.
3)The electrostatic field is along the displacement vector joining the source and the field point. The magnetic field is perpendicular to the plane containing the displacement vector r and the current element I dl.
4)There is an angle dependence in the Biot-Savart law which is not present in the electrostatic case.

r= (of moving charge particle having circular trajectory in magnetic field)/ in cyclotron

r= P/qB => mv/qB

T= ?n frequency=?(of moving charge particle having circular trajectory in magnetic field)/ in cyclotron

T= 2πm/qB
νc = qB/2πm

Kinetic energy of ion in cyclotron

K.E. = q^2B^2R^2/2m

time for one revolution of an ion is independent of its

speed or radius of its orbit.

Inside the dees the particles travel in a region.......electric field

Zero

frequency of revolution of the charged particle in a magnetic field is independent

Of its energy

Why cyclotron set up in vacuum

to minimise collisions between the ions and the air molecules

resonance condition in cyclotron

νa = νc
Frequency of both high frequency oscillator and charge particle are equal

. The increase in their kinetic energy is ......each time charge particle cross from one dee to another

qV

Velocity of charge particle when leaving cyclotron

V= qBR/m
R= radius of the trajectory at exit

When charge particle enters magnetic field such that its direction of velocity is perpendicular to the direction of magnetic field than it was in ...... motion

Circular

Time period is independent of

Radius of charged particle under influence of magnetic field

When charged particle goes undeflected under magnetic field

When it enters parallel or antiparallel to the magnetic field

MAGNETIC FIELD ON THE AXIS OF A CIRCULAR CURRENT LOOP, derive formula

B= μ0 i R^2 / 2(x^2 + R^2)^3/2
R= radius of loop
x= distance of point from axis of loop

MAGNETIC FIELD At the centre OF A CIRCULAR CURRENT LOOP

B= μ0 i/ 2R
R= radius of loop

Source of magnetic field

Current element

Whose unit is Tesla metre / ampere

Permeability

Magnetic field at any point on the axis of straight wire carrying current is

Zero

AMPERE'S CIRCUITAL LAW

§B.dl = μ0 i
Ampere's law states that this integral(over closed amperian loop) is equal to μ0 times the total current passing through the surface

Condition of "B" at amperian loop

(i) B is tangential to the loop and is a non-zero constant B, or
(ii) B is normal to the loop, or
(iii) Bvanishes.

Magnetic field hay distance "r" from it)due to straight infinite current-carrying wire

B×2πr = μ0 i
B=μ0 i /(2πr)

Magnetic field due to infinite wire "4" points

1)It implies that the field at every point on a circle of
radius r, (with the wire along the axis), is same in magnitude. In other words, the magnetic field possesses what is called a cylindrical symmetry. The field that normally can depend on three coordinates depends only on one: r.
2)The field direction at any point on this circle is tangential to it. Thus, the lines of constant magnitude of magnetic field form concentric circles
3)even though the wire is infinite, the field due to it at a non-zero distance is not infinite
4) right-hand rule(Grasp the wire in your right hand with your extended thumb pointing
in the direction of the current. Your fingers will curl around in the direction of the magnetic field.)

Amperian loop/ cylindrical symmetry

field at every point on a circle of radius r, (with the wire along the axis), is same in magnitude.

a long straight wire of a circular cross-section (radius a) carrying steady current I. The current I is uniformly distributed across this cross-section. Calculate the magnetic field in the region r < a and r > a.

Pg. 149

Long solenoid

solenoid's length is large compared to its radius

Wires used for solenoid

Enamelled wires are used for winding so that turns are insulated from each other.

Magnetic field outside solenoid

Zero

Magnetic field inside solenoid

(mid-point)uniform, strong and along the axis of the solenoid

Magnetic field due to infinite length solenoid

Bh=μ0I(nh)
B= μ0 n i
n= number of turns per unit length,

Magnetic field at (one of the )end due to infinite length solenoid

B= μ0 n i /2

Solenoid

Cylindrical coil of many tightly wound turns of insulated wire with generally diameter of coil smaller than its length is called solenoid

Toroid

Ring shaped closed solenoid

Derive formula for magnetic field of toroid

B= μ0N i/ 2πr
r be the average radius of the toroid and n be the number of turns per unit length.
B= μ0 n i (N = 2πr n )

the magnetic field in the open space inside (point P) and exterior to the toroid (point Q) is

Zero.
Pg.152

Parallel currents (a)..... and antiparallel currents (b)....

(a) attract
(b) repel

FORCE BETWEEN TWO PARALLEL CURRENTS, THE AMPERE

Fba = μ0 Ia Ib L / 2πd

Define 1 Ampere

The ampere is the value of that steady current which, when maintained in each of the two very long, straight, parallel conductors of negligible cross-section, and placed one metre apart in vacuum, would produce on each of these conductors a force equal to 2 × 10-7 newtons per metre of length.

current balance

measure the(two infinite || conductors) mechanical force.

TORQUE ON CURRENT LOOP, MAGNETIC DIPOLE
Torque on a rectangular current loop in a uniform magnetic field

τ = m×B
Or
τ = I A B sin θ
m= magnetic moment (direction by right hand rule)
B= external magnetic field
A= area of coil
θ= angle between mg ethic moment and external magnetic field

Condition for unstable and stable equilibrium for Torque on a rectangular current loop in a uniform magnetic field

Stable equilibrium - magnetic moment is parallel to external magnetic field
Unstable equilibrium- magnetic moment and external magnetic field are anti parallel

Magnetic moment

a measure of an object's tendency to align with a magnetic field
m=NIA

Whose dimensions are [A][L^2]

Magnetic moment, units Am^2

Analogy b/w magnetic dipole and electric dipole

B= μ0 2m/ 4π x^3
μ0 →1/ε0
m → pe (electrostatic dipole)
B → E
E= 2pe/ 4πε0 x^3
B for a point in the
plane of the loop
B= μ0 m/ 4π x^3

Magnetic field for a point in the
plane of the loop at a distance x from the centre

B= μ0 m/4π x^3

Difference between magnetic dipole and electric dipole

an electric dipole is built up of two elementary units — the charges (or electric monopoles). In magnetism, a magnetic dipole (or a current loop) is the most elementary element; magnetic monopoles, are not known to exist

Circular loop as magnetic dipole

Pg.160 , notes

gyromagnetic ratio

the ratio of the magnetic moment (field strength) of a rotating charged particle, such as an electron, to its angular momentum (frequency). The value of the gyromagnetic ratio of hydrogen is 42.57 MHz/Tesla

8.8 x 10^10 C/kg( for e-)

Bohr magneton

9.27 x 10^-24 Am^2
μl)min= e n h/4πme

Relation between The magnetic dipole moment and angular momentum of a revolving electron

μl=- e l/2m
The negative sign indicates that the angular momentum of the electron
is opposite in direction to the magnetic moment
μl/L = e/2m. -gyromagnetic ratio

Current sensitivity of galvanometer

It is defined as the deflection produced in the galvanometer per unit current flowing through it
S= NBA/C
OR
S= deflection angle/ current => NBA/C

Voltage sensitivity of galvanometer

The deflection produced in the galvanometer per unit voltage applied to it
S= NBA/ RC
C= Spring constant
R= resistance
B= external magnetic field A= area of coil
N= no. Of turns in coil

What's the use of radial magnetic field in MCG ( moving coil galvanometer)

This makes angle between magnetic moment and magnetic field always 90°, hence torque
It ensure that angle of deflection vary linearly with increase in curre

Why galvanometer cannot be used as ammeter

Galvanometer is a very sensitive device, it gives a full-scale deflection for a current of the order of μA.
For measuring currents, the galvanometer has to be connected in series, and as it has a large resistance, this will change the value of the current in the circuit. To overcome these difficulties, one attaches a small resistance rs, called shunt resistance, in parallel with the galvanometer coil; so that most of the current passes through the shunt.

Current sensitivity is independent of ......but voltage sensitivity depends on it

Resistance of coil

current sensitivity .....necessarily increase the voltage sensitivity

Not
Because voltage sensitivity also depends on resistance of coil
N → 2N
φ/ i → 2(φ/ i )
R →2R(the resistance of the galvanometer is also likely to double, since it is proportional to the length of the wire)
φ/ V → φ/ V

What kind of motion do we observe in a magnetic field?

There is no work done as the motion of the particle is perpendicular to the magnetic field.
No change in the magnitude of velocity, but the magnitude of momentum may change.

properties of the magnetic field

1. Defined at each point in space
2. Obeys the principle of superposition

Lorentz force

the summation of electric and magnetic forces at a point in space

properties of force due to a magnetic field

1. depends on q, v and B
2. F vanishes if v and B are parallel or antiparallel
3. F is perpendicular to the velocity and magnetic field

Unit of B

Newton Second per Coulomb meter. often written as Tesla. Gauss is used for everyday use since Tesla is too large

Force on a current carrying conductor

for a straight wire of current density n, length l, cross sectional area A, Magnetic field B and drift velocity V

What if V has a component along B?

The motion will be helical instead of circular.

radius of rotation

Angular frequency

pitch of the rotation

distance moved along the field in one rotation

Principle of a velocity selector

make qE=qvB, the velocity selector passes unaffected through the region where the two fields cross.

Working principle of a cyclotron

-Inside the dees, E does not act; B does
-they move in a circular path and leave the dees to be acted upon by E
-the E increases the energy of the particle as well as the radius of rotation
-leaves via a field of magnetic deflection.

time period and frequency of revolution in a cyclotron

resonance condition of a cyclotron

when the applied voltage is equal to the cyclotron frequency

velocity when the particles leave the dee

kinetic energy of the particles

basis of the operation of a cyclotron

the time for revolution of an ion is independent of its speed or radius of orbit