moving charges and magnetism

magnetic field

• it is the surrounding of a magnet/ current carrying conductor where magnetic influence can be experienced

• unit → T or Wb/m²

  • cgs unit → gauss (G) where 1T =10^4 G

• dimensions → [MT^-2A^-1]

oersted’s experiment

• it was conducted to prove that magnetic field is produced due to electric current

• he found that when he put a needle near a current carrying conductor, it gets deflected.

• the alignment of the deflected needle is tangential to an imaginary circle which has the straight wire as its centre and its plane perpendicular to the wire

• on increasing the current, the deflection increases

• also, he sprinkled iron filings around the wire, which arranged themselves in concentric circles with the wire

(draw the diagrams)

deflection in needle when theres no current, when there is current, and when the current is reversed

• no current: needle is parallel to wire

• yes current: the north pole gets deflected towards west

• reversed current: the north pole gets deflected towards east

(draw the diagrams)

ampere’s swimming rule

• it tells us how the needle deflects due to current

• SNOW - when the current flows from south to north, the north pole gets deflected towards west

biot-savart’s law

• it tells us the magnetic field induction at a point due to a small current element

• according to the law, dB at point P depends on,

  • dB \alpha I (current flowing thru conductor)

  • dB \alpha dl (length of the element)

  • dB \alpha sin\theta (sine of angle btw. length of element and line joining the element to point P)

  • dB \alpha 1/r² (square of distance btw. element and P)

• hence,

• dB = \muo/ 4\pi x Idlsin\theta/r²,

• where \muo = permeability of free space = 4\pi x 10^-7 Tm/A, dimensions → [MLT^-2A^-2]

biot-savart’s law in vector form

in vector form,

• dB = \muo/4\pi x Idl x r/ r³

• where, the direction of dB = direction of cross product dl x r

• here, dB is perpendicular to the plane containing dl and r, and is directed inwards

biot-savart’s law in terms of current density J

dB = \muo/ 4\pi (J x r/ r³) dV

(where J = I/A = Idl/ Adl = Idl/ dV)

biot-savart’s law in terms of charge and velocity

dB = \muo/4\pi q(v x r)/ r³

(since, Idl = qdl/ dt = qv)

biot-savart’s law in terms of magnetising intensity (H)

dH = dB/ \muo = 1/4\pi Idlsin\theta/r²

features of biot-savart’s law

• it is analogous to coulomb’s law

• the direction of dB is perpendicular to both Idl and r (right hand thumb rule)

• if \theta = 0, the point lies on the axis of the wire, and dB = 0

• if \theta = 90, the point lies perpendicular to the current element, and dB = max.

similarities between biot-savart’s law and coulomb’s law

• they are both long range, and depend inversely on square of distance from source to point

• the principle of superposition applies to both

• they’re both directly proportional to their sources, Idl and q respectively

differences between biot-savart’s law and coulomb’s law

• electric field is produced by a scalar source (q) while magnetic field is produced by a vector source (Idl)

• the electric field is along the displacement vector joining the source and point, while the magnetic field is perpendicular to the plane containing r and Idl

• theres an angle dependence in biot-savart’s law, while there isnt in coulomb’s law

relation between electric permittivity (\epsilono) and magnetic permeability (\muo)

• electric permittivity determines the degree of interaction of E with medium

• magnetic permeability measures the ability of the substance to acquire magnetisation in the field

\muo\epsilono = 1/c² , which is constant

(derive this)

magnetic field due to a current carrying conductor

• it is perpendicular to both dl and r

• B = \muoI/ 2\piR

• the direction can be determined by right hand thumb rule/ maxwell’s cork screw rule

magnetic field on the axis of a current carrying circular loop

• B = \muoIa²/ 2(r² + a²) ^3/2

• B = \muoNIa²/ 2(r² + a²) ^3/2 (for N turns)

• the direction of B is along the axis and away from the loop, when the current is in anti-clockwise direction

(derive this and draw the B-r variation graph)

magnetic field at the centre of a current carrying loop

• B = \muoI/ 2R

• for N turns, B = \muoNI / 2R

• for semicircle, B = \muoI/ 4R

• direction of B → from right hand rule

diagrams to show that a circular loop behaves as a magnet with 2 opposite poles

draw them

ampere’s circuital law

• the line integral of a magnetic field B around any closed path in vacuum is \muo times the net current I enclosed by the curve

• \intB.dl = \muoI

• its only applicable for an amperian loop

• the loop has to be such that either

  • B is tangential to the loop

  • B is normal to the loop

  • B vanishes

• it is similar to biot-savart’s law as they relate B and I, and express the same physical consequences of a steady current

derivation of magnetic field due to long straight thin wire using ampere’s law

derive this with diagram

we also get the following conclusions,

• B at every point on a circle of radius r is same → it is cylindrically symmetric

• direction of B at any