Various methods can be used to measure the mass of an atom. One possibility is through the use of a mass spectrometer. The basic feature of a Banbridge mass spectrometer is illustrated in figure. A particle carrying a charge +q is first sent through a velocity selector and comes out with velocity v = E/B.
The applied electric and magnetic fields satisfy the relation E = vB so that the trajectory of the particle is a straight line. Upon entering a region where a second magnetic field \(\vec{B}_{0}\) pointing into the page has been applied, the particle will move in a circular path with radius r and eventually strike the photographic plate.

(i) In mass spectrometer, the ions are sorted out in which of the following ways?

(ii) Radius of particle in second magnetic field B_o is

(iii) Which of the following will trace a circular trajectory wit largest radius?

(iv) Mass of the particle in terms q, B_o, B,r and E is

(v) The particle comes out of velocity selector along a straight line, because

Ampere's law gives a method to calculate the magnetic field due to given current distribution. According to it, the circulation \(\oint \vec{B} \cdot d \vec{l}\) of the resultant magnetic field along a closed plane curve is equal to \(\mu_{0}\) times the total current crossing the area bounded by the closed curve provided the electric field inside the loop remains constant. Ampere's law is more useful under certain symmetrical conditions. Consider one such case of a long Straight wire with circular cross-section (radius R) carrying current I uniformly distributed across this cross-section.

(i) The magnetic field at a radial distance r from the centre of the wire in the region r > R, is

(ii) The magnetic field at a distance r in the region r < R is

(iii) A long straight wire of a circular cross section (radius a) carries a steady current I and the current I is uniformly distributed across this cross-section. Which of the following plots represents the variation of magnitude of magnetic field B with distance r from the centre of the wire?

(iv) A long straight wire of radius R carries a steady current I. The current is uniformly distributed across its cross-section. The ratio of magnetic field at R/2 and 2R is

(v) A direct current I flows along the length of an infinitely long straight thin walled pipe, then the magnetic field is

The path of a charged particle in magnetic field depends upon angle between velocity and magnetic field.If velocity \(\vec{v}\) is at angle \(\theta\) to \(\vec{B}\) component of velocity parallel to magnetic field \((v \cos \theta)\) remains constant and component of velocity perpendicular to magnetic field \((v \sin \theta)\) is responsible for circular motion, thus the charge particle moves in a helical path.

The plane of the circle is perpendicular to the magnetic field and the axis of the helix is parallel to the magnetic field. The charged particle. moves along helical path touching the line parallel to the magnetic field passing through the starting point after each rotation.
Radius of circular path is \(r=\frac{m v \sin \theta}{1 v_{q} B}\)
Hence the resultant path of the charged particle will be a helix, with its axis along the direction of \(\vec{B}\) as shown in figure.
(i) When a positively charged particle enters into a uniform magnetic field with uniform velocity, its trajectory can be (i) a straight line (ii) a circle (iii) a helix.

(ii) Two charged particles A and B having the same charge, mass and speed enter into a magnetic field in such a way that the initial path of A makes an angle of 30° and that of B makes an angle of 90° with the field. Then the trajectory of

(iii) An electron having momentum 2.4 x 10^-23kg m/ s enters a region of uniform magnetic field of 0.15 T. The field vector makes an angle of 30° with the initial velocity vector of the electron. The radius of the helical path of the electron in the field shall be

(iv) The magnetic field in a certain region of space is given by \(\vec{B}=8.35 \times 10^{-2} \hat{i}\) T. A proton is shot into the field with velocity \(\vec{v}=\left(2 \times 10^{5} \hat{i}+4 \times 10^{5} \hat{j}\right) \mathrm{m} / \mathrm{s}\) The proton follows a helical path in the field. The distance moved by proton in the x-direction during the period of one revolution in the yz-plane will be
(Mass of proton = 1.67 x 10^-27kg)

(v) The frequency of revolution of the particle is

An electron with speed V_o << c moves in a circle ofradius r_o in a uniform magnetic field. This electron is able to traverse a circular path as magnetic field is perpendicular to the velocity of the electron. A force acts on the particle perpendicular to both \(\vec{v}_{0}\) and \(\vec{B}\). This force continuously deflects the particle sideways without  changing its speed and the particle will move along a circle perpendicular to the field. The time required for one revolution of the electron is T_o .

(i) If the speed of the electron is now doubled to 2v_o.The radius of the circle will change to

(ii) If v_o = 2v_o then the time required for one revolution of the electron will change to

(iii) A charged particles is projected in a magnetic field \(\vec{B}=(2 \hat{i}+4 \hat{j}) \times 10^{2} \mathrm{~T}\) The acceleration of the particle is found to be \(\vec{a}=(x \hat{i}+2 \hat{j}) \mathrm{m} \mathrm{s}^{-2}\). Find the value of x.

(iv) If the given electron has a velocity not perpendicular to B, then trajectory of the electron is

(v) If this electron of charge (e) is moving parallel to uniform magnetic field with constant velocity v, the force acting on the electron is

When a rectangular loop PQRS of sides 'a' and 'b' carrying current I is placed in uniform magnetic field \(\vec{B}\) such that area vector \(\vec{A}\) makes an angle \(\theta\) with direction of magnetic field, then forces on the arms QR and SP of loop are equal, opposite and collinear, thereby perfectly cancel each other, whereas forces on the arms PQ and RS of loop are equal and opposite but not collinear, so they give rise to torque on the loop.

Force on side PQ or RS ofloop is F = IbB sin 90° = Ib B and perpendicular distance between two non-collinear forces is \(r_{\perp}\) = a sin \(\theta\)

So, torque on the loop, \(\tau=I A B \sin \theta\)
In vector form torque \(\vec{\tau}=\vec{M} \times \vec{B}\)
where \(\vec{M}=N I \vec{A}\) is called magnetic dipole moment of current loop and is directed in direction of area vector \(\vec{A}\) i.e., normal to the plane ofloop.
(i) A circular loop of area 1 cm², carrying a current of 10 A is placed in a magnetic field of 0.1T perpendicular to the plane of the loop. The torque on the loop due to the magnetic field is

(ii) Relation between magnetic moment and angular velocity is

(ill) A current loop in a magnetic field

(iv) The magnetic moment of a current I carrying circular coil of radius r and number of turns N varies as

(v) A rectangular coil carrying current is placed in a non-uniform magnetic field. On that coil the total

(i) (c): In mass spectrometer, the ions are sorted out by accelerating them through electric and magnetic field.

A magnetic field can be produced by moving, charges or electric currents. The basic equation governing the magnetic field due to a current distribution is the Biot-Savart law.
Finding the magnetic field resulting from a current distribution involves the vector product, and is inherently a calculas problem when the distance from the current to the field point is continuously changing.
According to this law, the magnetic field at a point due to a current element oflength \(d \vec{l}\) carrying current I, at a distance r from the element is \(d B=\frac{\mu_{0}}{4 \pi} \frac{I(d \vec{l} \times \vec{r})}{r^{3}}\)
Biot -Savart law has certain similarities as well as difference with Coloumbs law for electrostatic field e.g., there is an angle dependence in Biot-Savart law which is not present in electrostatic case.
(i) The direction of magnetic field \(d \vec{B}\) due to a current element \(I d \vec{l}\) at a point of distance \(\vec{r}\) from it, when a current I passes through a long conductor is in the direction

(ii) The magnetic field due to a current in a straight wire segment of length L at a point on its perpendicular bisector at a distance r (r >> L)

(iii) Two long straight wires are set parallel to each other. Each carries a current i in the same direction and the separation between them is 2r. The intensity of the magnetic field midway t between them is

(iv) A long straight wire carries a current along the z-axis for any two points in the x - y plane. Which of the following is always false?

(v) Biot-Savart law can be expressed alternatively as

Moving coil galvanometer operates on Permanent Magnet Moving Coil (PMMC) mechanism and was designed by the scientist D'arsonval.
Moving coil galvanometers are of two types
(i) Suspended coil
(ii) Pivoted coil type or tangent galvanometer.
Its working is based on the fact that when a current carrying coil is placed in a magnetic field, it experiences a torque. This torque tends to rotate the coil about its axis of suspension in such a way that the magnetic flux passing through the coil is maximum.

(i) A moving coil galvanometer is an instrument which

(ii) To make the field radial in a moving coil galvanometer

(iii) The deflection in a moving coil galvanometer is

(iv) In a moving coil galvanometer, having a coil of N-turns of area A and carrying current I is placed in a radial field of strength B.
The torque acting on the coil is

(v) To increase the current sensitivity of a moving coil galvanometer, we should decrease

A galvanometer can be converted into voltmeter of given range by connecting a suitable resistance R_s in series with the galvanometer, whose value is given by
\(R_{s}=\frac{V}{I_{g}}-G\)
where V is the voltage to be measured, I_g is the current for full scale deflection of galvanometer and G is the resistance of galvanometer

Series resistort (R_s) increases range of voltmeter and the effective resistance of galvanometer. It also protects the galvanometer from damage due to large current.
Voltmeter is a high resistance instrument and it is always connected in parallel with the circuit element across which potential difference is to be measured. An ideal voltmeter has infinite resistance
In order to increase the range of voltmeter n times the value of resistance to be connected in series with galvanometer is R_s = (n - l)G.
(I) 10 mA current can pass through a galvanometer of resistance \(25 \Omega\) What resistance in series should be connected through it, so that it is converted into a voltmeter of 100 V?

(ii) There are 3 voltmeter A, B, C having the same range but their resistance are \(15,000 \Omega, 10,000 \Omega\) and \(5,000 \Omega\) respectively. The best voltmeter amongst them is the one whose resistance is

(iii) A milliammeter of range 0 to 25 mA and resistance of \(10 \Omega\) is to be converted into a voltmeter with a range of 0 to 25 V. The resistance that should be connected in series will be

(iv) To convert a moving coil galvanometer (MCG) into a voltmeter

(v) The resistance of an ideal voltmeter is

A charged particle moving in a magnetic field experiences a force that is proportional to the strength of the magnetic field, the component of the velocity that is perpendicular to the magnetic field and the charge of the particle.
This force is given by \(\vec{F}=q(\vec{v} \times \vec{B})\) where q is the electric charge of the particle, v is the instantaneous velocity of the particle, and B is the magnetic field (in tesla).
The direction of force is determined by the rules of cross product of two vectors
Force is perpendicular to both velocity and magnetic field. Its direction is same as \(\vec{v} \times \vec{B}\) if q is positive and opposite of \(\vec{v} \times \vec{B}\) if q is negative
The force is always perpendicular to both the velocity of the particle and the magnetic field that created it. Because the magnetic force is always perpendicular to the motion, the magnetic field can do no work on an isolated charge. It can only do work indirectly, via the electric field generated by a changing magnetic field.

(I) When a magnetic field is applied on a stationary electron, it

(ii) A proton is projected with a uniform velocity v along the axis of a current carrying solenoid, then

(iii) A charged particle experiences magnetic force in the presence of magnetic field. Which of the following statement is correct?

(iv) A charge q moves with a velocity 2 ms^-1 along x-axis in a uniform magnetic field \(\vec{B}=(\hat{i}+2 \hat{j}+3 \hat{k}) \mathrm{T}\) then charge will experience a force

(v) Moving charge will produce

A solenoid is a long coil of wire tightly wound in the helical form. Solenoid consists of closely stacked rings electrically insulated from each other wrapped around a non-conducting cylinder.
Figure below shows the magnetic field lines of a solenoid carrying a steady current 1.We see that if the turns are closely spaced, the resulting magnetic field inside the solenoid becomes fairly uniform, provided that the length of the solenoid is much greater than its diameter, For an "ideal" solenoid, which is infinitely long with turns tightly packed, the magnetic field inside the solenoid is uniform and parallel to the axis, and vanishes outside the solenoid.

(i) Along solenoid has 800 turns per metre length of solenoid. A current of 1.6A flows through it. The magnetic induction at the end of the solenoid on its axis is

(ii) Choose the correct statement in the following

(iii) The magnetic field (B) inside a long solenoid having n turns per unit length and carrying current I when iron core is kept in it is \(\left(\mu_{0}=\text { permeability of vacuum, } \chi=\right.\text { magnetic susceptibility) }\)

(iv) A solenoid oflength I and having n turns carries a current I is in anticlockwise direction. The magnetic field is

(v) The magnitude of the magnetic field inside a long solenoid is increased by

Electric fields interact with both stationary and moving charges while magnetic fields affect only moving charges. Therefore, when a charged particle of charge q and mass m moving with velocity 'v' enters a region where both eletric \((\vec{E})\) and magnetic \((\vec{B})\) fields are simultaneously applied, then this charge particle will experience a force called Lorentz force given by 
\(\vec{F}=q \vec{E}+q(\vec{v} \times \vec{B})\)
and the motion of the charged particle is quite complicated. But by adjusting the value of \((\vec{E})\) and \((\vec{B})\) the velocity of.charge particle can be controlled.
(i) The figure below shows three situation in which a charged particle with velocity \(\vec{v}\) travels through a uniform magnetic field \(\overrightarrow{\boldsymbol{B}}\). In each situation, what is the direction of the magnetic force \(\overrightarrow{F_{B}}\) on the particle? Also, draw the trajectory of particle in each case.


(ii) In the adjoining figure four directions for the velocity vector \(\vec{v}\)of a positively charged particle moving through a uniform electric field \(\overrightarrow{\boldsymbol{E}}\) (directed into the page and represent by \(\otimes \text { ) }\)and a unform magnetic field \(\overrightarrow{\boldsymbol{B}}\) .

(a) Rank directions 1, 2 and 3 according to the magnitude of the net force on the particle, greatest first.
(b) Among all four directions, which might result in a net force of zero?

Figure shows a rectangular coil ABCD placed between magnetic N-pole and S-pole.
There is a current in the coil

An upward force F = 9.6 x 10^-3 N is acting on the side AB of the coil.
(i) (a) Explain why there is a force on side AB.
(b) Determine the direction of current in AB and state how the direction is deduced.
(ii) Both sides AB and CD of the coil are at a distance 2 cm from the axis. Determine the moment acting on the coil ABCD.
(iii) For which purpose above arrangement can be used?
In which device above arrangement is used?

A loop of flexible conducting wire of length 31.4 cm carrying current of 1.5 A is placed in a magnetic field as shown. If current flowing through the wire in anticlockwise direction is switched on then what will be

(i) the shape of the loop?
(ii) determine net magnetic field at the centre of the loop. If it is given that current carrying loop is placed in a region having external magnetic field of 3.6 G.

What is the force experienced by a semi-circular wire of radius R carrying current 'I' and is placed in a region where uniform magnetic flux density (B) is as shown in the figures (a), (b) and (c) respectively?
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If a straight current carrying conductor is placed as shown. Points A and B lies on west and east side of a conductor at a distance x each from the conductor. Then relate the magnetic field at A to that at B. Also draw the corresponding magnetic field lines.