Moving Charges and Magnetism

1. Earlier electricity and magnetism were considered two separate domains of Physics. However, on the basis of Oersted’s experiment and subsequent work it has been established that moving charges or currents produce a magnetic field in the surrounding space. Subsequently on the basis of more intense experimentation unified basic laws of electromagnetism were developed, which led to the discovery of electromagnetic waves.

2. Magnetic field is characterised by a magnetic field vector (also known as magnetic induction or magnetic flux density) $ \vec{B} $. SI unit of magnetic field $ \vec{B} $ is 1 tesla (1 T). Sometimes it is also referred as weber/m² (Wb m² ). C.G.S. unit of  $ \vec{B} $ is 1 gauss (1 G), where 1 G = 10^-4 T.

3. Due to a straight conductor, the magnetic field lines formed are concentric circles around the conductor with the conducting wire at the centre. The magnetic line formed closed loops with no beginning and no end.

4. The direction of magnetic field due to a straight, current carrying conductor is given by right hand thumb rule. According to this rule, grasp the conductor carrying current in your right hand with the thumb perpendicular to the fingers and pointing in the direction of the current, then the curls of the figures point in the direction of magnetic field $ \vec{B} $ associated with the conductor.

5. According to Biot-savart law, the magnetic field $\vec{dB} $ in free space at a point P at a distance ‘r’ from a differential current element $ I \vec{dl} $ is given by \[\vec{dB} = \frac{\mu_0}{4\pi} \frac{I\vec{dl}x \vec{r}}{r^3}\]Direction of $ \vec{dB} $ is that of  $ \vec{dl}* \vec{r}$ i.e., direction of magnetic field is perpendicular to both $\vec{dl} $ and $ \vec{r} $. Magnitude of magnetic field is given by \[dB = \frac{\mu_0}{4\pi} \frac{Idl sin \theta}{r^2}\]

6. The term μ₀ appearing in the Biot-Savart law is known as the magnetic permeability of free space. Value of μ₀ = 4π * 10^-7 tesla metre / ampere (T m A^-1).

7. For a straight current carrying thin wire of finite length the magnetic field at a point situated at a normal distance R from the centre of conductor is given by \[B = \frac{\mu_0 I}{4\pi R} [ sin \phi_1 + sin \phi_2]\] where I is the current flowing in the wire and $ \phi_1 $ and $ \phi_2 $ are the angles by the two ends of given conductor from normal direction. 

8. For a current carrying thin wire of infinite length $ \phi_1 = \phi_2  = 90 $ and hence \[B = \frac{\mu_0 I}{2\pi R}\]However, for a point P lying on the thin wire itself, the magnetic field B is zero. Magnetic field at a point situated near one end of an infinitely long current carrying wire is given by \[B =\frac{\mu_0 I}{4\pi R}\]

9. For a circular wire loop of radius R and carrying a current I the magnetic field at the centre point of circle is given by \[B = \frac{\mu_0I}{2R}\]Direction of magnetic field due to a circular loop carrying current is given by right hand palm rule. According to it, curl the palm of your right hand around the circular wire with the fingers pointing in the direction of the current. Then the thumb of right hand gives the direction of the magnetic field. If we have a circular coil of N turns then the magnetic field at its centre is given by \[B = \frac{\mu_0 N I}{2R}\] 

10. Magnetic field B due to a current I in a circular coil of N turns each of radius R at a point P on its axial line at a distance x from the centre of coil is given by \[B = \frac{\mu_0 N I R^2}{2 (R^2 + x^2 )^{3/2} }\]

The above expression reduces to the following form : 

(i) At the centre of circular coil (i.e., x = 0),  \[B = \frac{\mu_0N I}{ 2R }\] 

(ii) At points far away from the centre i.e., x >> R, \[B = \frac{\mu_0 N I R^2}{ 2x^3} = \frac{\mu_0}{4\pi}\frac{ 2 I A}{x^3}\] where  A = π R² = area of circular loop. The magnetic field is directed axially in the direction given by right hand rule. 

11. If an open surface is bounded by a loop then Ampere’s circuital law states that the line integral of the magnetic field along the loop is equal to μ₀ times the net current passing through the surface. By the net current we mean the algebraic sum of the current within that loop i.e., where I_e is the net current enclosed. While applying Ampere’s law we follow the right hand rule. Let the finger of the right hand be curled in the sense the boundary is traversed in the loop along the direction of integral, then the direction of the thumb gives the sense in which the current I is regarded as positive.

12. While applying Ampere’s circuital law, if we choose the amperian loop such that at each point of the loop either $ \vec{B} $ is tangential to the loop and is a non-zero constant B or $ \vec{B} $ is normal to the loop, then \[\oint \vec{B}.\vec{dl} = BL = \mu_0 I_e\]Here L is the length of the loop for which B is tangential.

13. The simplest application of Ampere’s circuital law is to determine magnetic field at a point situated at a normal distance R from a long current carrying straight wire of a circular cross-section. From this we find that  

(i)\[B = \frac{\mu_0 I}{2\pi R}\]where R >= r (r is the radius of the wire) and

(ii) \[B = \frac{\mu_0 I}{2\pi r^2}R\]when  R <= r

14. A straight solenoid is prepared by a long, insulted copper wire wound in the form of a helix with neighbouring turns very closely spaced. If the length of the solenoid is large enough as compared to its transverse cross-section, the solenoid is considered to be a long, straight solenoid. In a solenoid magnetic field due to all the turns is in the same direction and being added up. The net magnetic field inside a tightly wound infinite solenoid is uniform and axial but zero outside the solenoid. Magnetic field inside the infinitely, long solenoid carrying current I by applying Ampere's Circuital Law is given by \[B = \mu_0 nI = \frac{\mu_0 NI}{l}\]Where n = number of turns per unit length =$ \frac{N}{L} $. The direction of the magnetic field is given by right hand palm rule. Magnetic field due to an infinitely long solenoid is along the axis of the solenoid. At a point just near the free end of a long, straight solenoid the magnetic field \[B = \frac{μ0 n I}{2}\]

15. A toroid is a hollow circular ring on which a large number of turns of an insulted copper wire are closely wound. For a toroidal solenoid carrying current the magnetic field at any point : (i) outside the toroid, (ii) inside the open space in the toroid is zero. Magnetic field inside a toroidal solenoid of radius R and having N turns in all and carrying a current I is given by \[B = \frac{\mu_0 NI}{2\pi R} = \mu_0 n I\]where $ n = \frac{N}{2\pi R} $ is the number of turns per unit length.

16. The magnetic field inside a hollow pipe (or tube ) of current is zero. 

17. Magnetic field B at the centre due to a current flowing in a circular arc shaped conductor is  $ \frac{θ}{2π} $ times the magnetic field due to a circular loop, where θ is the angle subtended by the arc at the centre. Thus, \[B = \frac{\theta}{2\pi} \frac{\mu_0 I}{2R} = \frac{l}{2\pi R}\frac{\mu_0 I}{2R} = \frac{\mu_0 Il}{4\pi R^2}\] Where l is the length of conducting arc.

18. The force $\vec{F_B}$  acting on a electric charge q moving in a magnetic field $\vec{B}$ with the velocity $\vec{v}$ is called the magnetic Lorentz force and is given by \[\vec{F} = q (\vec{v}*\vec{B} )\]or \[F = qvBsin\theta\]where θ is the angle between the direction of v and the magnetic field B as given by right hand rule. Lorentz force is non-conservative force.

19. Magnitude of Lorentz magnetic force is determined by the component of velocity of direction perpendicular to that of magnetic field B. 

(i)if angle θ between B and v is 0⁰ or 180⁰ i.e., charged particle in moving parallel or antiparallel to the magnetic field, force F_B = 0.

(ii) if motion of charged particle in a direction perpendicular to that of magnetic field, force acting, on its maximum having a value F_B = q v B.

20. The charged particle entering a uniform magnetic field B, in a direction perpendicular to that of B, with a velocity v moves along a circular path of radius ‘r’ in a plane at right angle to B given by \[r = \frac{mv}{qB} = \frac{p}{qB} = \frac{\sqrt{2mK}}{qB}\]Where p = momentum of charged particle and K its kinetic energy. 

21. The magnetic force behave as the centripetal force and does not do any work. Thus, it does not change the kenetic energy of moving charge. However, due to change in direction of motion the velocity and momenta of charged particle change. The time period is complete one revolution in the circular trajectory is \[T = \frac{2\pi m}{qB}\]and the frequency of revolution is \[\nu = \frac{1}{T}=\frac{qB}{2\pi m}\]The frequency is independent of the charged particle’s speed as well as radius.

22. A charged particle entering a uniform magnetic field B in a direction making an angle θ from B, describes a helical path of radius r given by \[r = \frac{mvsin\theta }{qB}\]The pitch p of a helical path (I.e., linear distance covered in one revolution along the direction of B ) is given by \[p = \frac{2\pi mvcos\theta }{qB}\]

23. When a charged particle q moves along the direction of an electric field $ \vec{E} $, its motion is accelerated or retarded depending on the sign of charge q. However, the path of charged particle remain a straight line. When a charged particle is allowed to entre an electric field in a direction perpendicular to that of electric field, path of the particle is a parabolic path.

24. A charge q moving simultaneously in an electric and a magnetic field experience a force called total Lorentz force F, given by \[\vec{F} =\vec{F_B} + \vec{F_E} = q(\vec{v}*\vec{B}) + q\vec{E} = q[(\vec{v}*\vec{B}) + \vec{E}]\]If electric field $ \vec{E} $ and magnetic field $ \vec{B} $ are mutually perpendicular to each other as well as perpendicular to $ \vec{v} $ then $\vec{F_B}$ and $\vec{F_E}$ are along same straight line. If we adjust values of E and B such that magnitudes of F_B and F_E are equal and opposite, then net force on charge is zero and the charged particle goes undeviated. It happens, when \[qvB = qE \implies v = \frac{E}{B}\] 

25. A cyclotron uses both electric and magnetic fields, mutually crossed one, to accelerate charged particles or ions to high energies. Under a magnetic field the charged particle describes circular paths but after every half revolution it is suitably accelerated by the oscillating electric field operating at cyclotron frequency, whose value is given by \[\nu_c = \frac{qB}{2\pi m}\]The maximum K.E. of ion beam obtained from a cyclotron is \[K_{max} = \frac{1}{2}mv_{max}^2 = \frac{q^2B^2R^2}{2m}\]where R is the radius of the Dees of the cyclotron.

26. A current element $ I \vec{dl} $ when placed a uniform magnetic field $ \vec{B} $ experience a mechanical force \[\vec{F} = I(\vec{dl} * \vec{B} )\]For a straight linear conductor \[\vec{F} = I(\vec{l}*\vec{B})\]and \[F = IlBsin\theta\]The force $ \vec{F} $ is in a plane perpendicular to both $ \vec{dl} $ (or $\vec{I} $ ) and $\vec{ B} $ in the direction of their cross product. Direction of mechanical force acting on a current carrying plane conductor can be noted with the help of Fleming’s left hand rule. According to it stretch out the central finger, forefinger and thumb of your left hand to be mutually perpendicular to each other. If centre finger points in the direction of current and the forefinger in the direction of magnetic field then the thumb will point in the direction of the force.

27. Two parallel, straight, long, current carrying conductors attract each other if the current flowing in them in the same direction but repel each other if the currents are in mutually opposite directions. The force of interaction per unit length is \[\frac{F}{l} = \frac{\mu_0}{4\pi} \frac{2I_1I_2}{d}\]where d is the separation between the wires. 

28. A current flowing in a closed loop (either circular or of any other shape) produces a magnetic field pattern like that produced by a magnetic dipole. The magnetic moment $\vec{m} $ of a current loop is given by \[\vec{m} = I\vec{A}\]and for a coil of N turns \[\vec{m} = N I \vec{A}\]where A = area vector of the loop and NI = number of ampere turns in the coil. The direction of magnetic moment is given by right hand rule. SI unit of magnetic dipole moment in ampere meter² (A m²).

29. A current loop produced a magnetic field and behaves like a magnetic dipole at large distances. Moreover, a current loop is subject to torque like a magnetic needle, hence we conclude that ordinarily all magnetism is due to circulating current. When a coil carrying current of magnetic dipole moment \[\vec{m} = N I \vec{A}\]is placed in an orientation θ with a uniform magnetic field B , it experiences a torque given by \[\vec{\tau} = \vec{m}*{\vec{B}} , \tau = mBsin\theta = NIABsin\theta\]For a radical magnetic field, when magnetic field lines are perpendicular to the arms of a rectangular coil in every orientation of the coil, θ = 90⁰ and, hence \[\tau = mB = NAIB\]

30. A moving coil galvanometer is an extremely sensitive device and measure electric current flowing in a current when placed in a radial magnetic field. Current can be measured by using the formula \[I = \frac{k}{NAB}\phi\]where k = torsional constant of the suspension fibre (or spring) of galvanometer and Ф = angular deflection of the coil on passing current through it. The term $ \frac{Ф}{I} $ i.e., deflection per unit current is known as the current sensitivity of a galvanometer. Current sensitivity \[\frac{\phi}{I} = \frac{N A B}{k}\]Thus, to enhance the current sensitivity we use a strong magnetic field B and suspension fibre (or spring ) of small value of torsional constant. However, increase in N and A is not possible beyond a limit due to practical problems.

31. A galvanometer can be converted into an ammeter of an appropriate range by connecting a suitable, small shunt resistance in parallel to the galvanometer. If a galvanometer having resistance R_G  and giving full scale deflection for a current I_g is to be converted into an ammeter of range I ampere, then shunt resiistance $r_s$ used in parallel is given by \[r_s = \frac{R_G I}{I-I_g}\]The net resistance of ammeter is \[\frac{R_G* r_s}{R_G + r_s }\]which is extremely small. Ammeter is always joined in series of the electrical curcuit in which is to be measured.

32. A galvanometer may be converted into a voltmeter of given range V by joining a suitable high resistance R in series of galvanometer, such that \[R = \frac{V}{I_g} - R_G\]The net resistance of a voltmeter is (R + R _G), which is quite high. A voltmeter is always connected in parallel of an electrical circuit across the points, potential difference between which is to be measured.

33. A revolving electron in an orbit around a nucleus constitutes a current and there will be an orbital magnetic moment $\mu_l$ having a magnitude given by \[\mu_l = \frac{n e h}{4\pi m_e }\]where n = 1,2, 3,…etc. In vector notations, we have \[\vec{\mu_l} = -\frac{ e }{2 me }\vec{l}\]where $\vec{l} $ is the orbital angular momentum of electron $( l = m_evr) $. The negative sign indicates that the angular momentum of the electron is opposite in direction to orbital magnetic moment.

34. The minimum value of orbital magnetic moment of a revolving electron is given by \[\mu_{min} = \frac{e h }{4\pi m_e }\] (when n = 1) and has a value 9.27 x 10^-24 A m². This term \[\mu_{min} = \frac{e h}{4\pi m_e } = 9.27 *10^{-24} A m^2\]is known as Bohr magneton. Beside the orbital moment, the electron has a intrinsic magnetic moment. It is called the spin magnetic moment.  

    

Note:  "*" indicate vector product of two physical quantities.