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Magnetic Effects Of Current And Magnetism

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Magnetic Effects of Current and Magnetism PDF Notes, Important Questions and Synopsis

SYNOPSIS

  • Lorentz force: Force on a charge q moving with velocity v in the presence of magnetic and electric fields B and E.begin mathsize 12px style straight F with rightwards arrow on top equals straight q left parenthesis straight v with rightwards arrow on top straight x straight B with rightwards arrow on top plus straight E with rightwards arrow on top right parenthesis end style
  • The magnetic force begin mathsize 12px style straight F with rightwards arrow on top subscript straight B equals straight q left parenthesis straight v with rightwards arrow on top straight x straight B with rightwards arrow on top right parenthesis end styleis normal to v with rightwards arrow on top and work done by it is zero.
  • Force F on a straight conductor of length 𝓁 and carrying a steady current I placed in a uniform external magnetic field B, begin mathsize 12px style straight F with rightwards arrow on top equals straight I calligraphic l with rightwards arrow on top straight x straight B with rightwards arrow on top end style In a uniform magnetic field the force, dF = IBdl Sinθ, does not depend on the position vector r of the current element.Thus, this force is non-central. The force begin mathsize 12px style straight d straight F with rightwards arrow on top end style is always perpendicular to the plane containing begin mathsize 12px style straight B with rightwards arrow on top    and    dl with rightwards arrow on top end style
  • Biot–Savart law asserts that the magnetic field begin mathsize 12px style straight d straight B with rightwards arrow on top end style due to an element begin mathsize 12px style straight d calligraphic l with rightwards arrow on top end style carrying a steady current I at a point P at a distance r from the current element is

    begin mathsize 12px style straight d straight B with rightwards arrow on top equals fraction numerator straight mu subscript 0 over denominator 4 straight pi end fraction straight I fraction numerator straight d calligraphic l with rightwards arrow on top cross times text    end text straight r with rightwards arrow on top over denominator straight r cubed end fraction end style
  • The magnetic field due to a circular coil of radius R carrying a current I at an axial distance x from the centre is 

    begin mathsize 12px style straight B equals fraction numerator straight mu subscript 0 IR squared over denominator 2 open parentheses straight x squared plus straight R squared close parentheses to the power of 3 divided by 2 end exponent end fraction end style

    At the centre of the coil,

    begin mathsize 12px style straight B equals fraction numerator straight mu subscript 0 straight I over denominator 2 straight R end fraction end style

  • Ampere’s circuital law: For an open surface S bounded by a loop C,

    begin mathsize 12px style contour integral for straight c of straight B with rightwards arrow on top. straight d calligraphic l with rightwards arrow on top equals straight mu subscript 0 straight I end style ,
    where I refers to the current passing through S.

    If B is directed along the tangent to every point on the perimeter, then

    begin mathsize 12px style BL equals straight mu subscript 0 straight I subscript straight e end style

    Where Ie is the net current enclosed by the closed circuit. Ampere’s law is an important tool in calculating the magnetic field due to current distribution. However, this usefulness is limited to only a few cases where the magnetic field has a symmetrical distribution in space. For example, this law cannot be used to find the magnetic field at the centre of a current-carrying loop. Ampere’s circuital law is not an independent law, but it is derived from Biot–Savart law

  • The magnetic field at a distance R from a long, straight wire carrying a current I is given by

    begin mathsize 12px style straight B equals fraction numerator straight mu subscript 0 straight I over denominator 2 text   end text straight R end fraction end style
    The field lines are circles concentric with the wire.

  • Magnetic field B inside a long solenoid carrying a current I is

    B = μ0 nI
    where n is the number of turns per unit length.
    For a toriod,

    begin mathsize 12px style straight B equals fraction numerator straight mu subscript 0 NI over denominator 2 πr end fraction end style

    where N is the total number of turns and r is the average radius.

  • Magnetic moment m of a planar loop carrying a current I with N closely wound turns and an area A is
    begin mathsize 12px style straight m with rightwards arrow on top equals NI straight A with rightwards arrow on top end style
    Direction of begin mathsize 12px style straight m with rightwards arrow on top end style is given by the right-hand thumb rule.
    Right-hand thumb rule: Curl the palm of your right hand along the loop with the fingers pointing in the direction of the current. The thumb sticking out gives the direction of begin mathsize 12px style straight m with rightwards arrow on top end style (and begin mathsize 12px style straight A with rightwards arrow on top end style ).
    When this loop is placed in a uniform magnetic field B, the force F on it is F = 0
    and the torque on it is begin mathsize 12px style straight tau with rightwards arrow on top equals straight m with rightwards arrow on top straight x straight B with rightwards arrow on top end style
    In a moving coil galvanometer, this torque is balanced by a counter torque due to a spring yielding
    kϕ = NI AB
    Where ϕ is the equilibrium deflection and k is the torsion constant of the spring.


Uses of a moving coil galvanometer:

  • It is used to detect electric current in a circuit, e.g. Wheatstone Bridge.
  • It is converted to an ammeter by putting a small resistance parallel to it.
  • It is used as an ohmmeter.
  • An electron moving around the central nucleus has a magnetic moment begin mathsize 12px style straight mu subscript calligraphic l end style given by
     begin mathsize 12px style straight mu subscript calligraphic l equals fraction numerator straight e over denominator 2 straight m end fraction calligraphic l end style

    Where 𝓁 is the magnitude of the angular momentum of the circulating electron about the central nucleus. The smallest value of μ𝓁 is called the Bohr magneton μB, and it is μB = 9.27 × 10–24 J/T.

  • Cyclotron:

    A cyclotron is a device used to accelerate positively charged particles (like protons, α-particles, deuterons, ions etc.) to acquire enough energy to carry out nuclear disintegration.
    A charge q executes a circular motion with frequency called the cyclotron frequency given by
    begin mathsize 12px style straight nu subscript straight c equals fraction numerator qB over denominator 2 πm end fraction end style

    This cyclotron frequency is independent of the particle’s speed and radius. 
    Time period
    begin mathsize 12px style straight T equals fraction numerator 2 πm over denominator qB end fraction end style

    Radius
    begin mathsize 12px style table attributes columnalign left end attributes row cell qvB equals mv squared over straight r end cell row cell straight r equals mv over qB end cell end table end style

    Limitations of a cyclotron:

    A cyclotron cannot accelerate uncharged particles like neutrons. Positively charged particles with large mass (i.e. ions) cannot be accelerated after a certain speed in the cyclotron.

  • Magnetic materials tend to point in the north–south direction.
    Like magnetic poles repel and unlike poles attract each other.
    Cutting a bar magnet in two leads to two smaller magnets. 
    Magnetic poles cannot be isolated.

  • When a bar magnet of dipole moment begin mathsize 12px style stack text m end text with rightwards arrow on top end style is placed in a uniform magnetic field begin mathsize 12px style stack text B end text with rightwards arrow on top end style,

  • The force on it is zero.
  • The torque on it is begin mathsize 12px style straight m with rightwards arrow on top space straight x space straight B with rightwards arrow on top end style.
  • Its potential energy is begin mathsize 12px style -  straight m with rightwards arrow on top .  straight B with rightwards arrow on top end style, where we choose the zero of the energy at the orientation when is perpendicular to .
  • Consider a bar magnet of size 𝓁 nd magnetic moment begin mathsize 12px style stack text m end text with rightwards arrow on top end style at a distance r from its midpoint, where r >>𝓁; the magnetic field begin mathsize 12px style stack text B end text with rightwards arrow on top end style due to this bar is

    begin mathsize 12px style table attributes columnalign left end attributes row cell straight B with rightwards arrow on top equals fraction numerator µ subscript 0 straight m with rightwards arrow on top over denominator 2 πr cubed end fraction text                  end text open parentheses along text    end text axis close parentheses end cell row cell text      end text equals negative fraction numerator µ subscript 0 straight m with rightwards arrow on top over denominator 4 πr cubed end fraction text              end text open parentheses along text    end text equator close parentheses end cell end table end style

  • Gauss’s law for magnetism: The net magnetic flux through any closed surface is zero.

    begin mathsize 12px style blank to the power of straight ϕ subscript straight capital beta equals end exponent sum for table attributes columnalign left end attributes row cell all text end text area end cell row cell elements text   end text straight capital delta straight S with rightwards arrow on top end cell end table of straight B with rightwards arrow on top. straight capital delta straight S with rightwards arrow on top equals 0 end style

  • Curie’s law: According to Curie’s law, the susceptibility of a paramagnetic substance is inversely proportional to the absolute temperature: m = c/T, where c is a constant called the Curie constant.

  • The pole near the geographic north pole of the Earth is called the north magnetic pole. 
    The pole near the geographic south pole is called the south magnetic pole. 
    The magnitude of the magnetic field on the Earth’s surface = 4 × 10−5 T.

  • Three quantities are needed to specify the magnetic field of the Earth on its surface—the horizontal component, the magnetic declination and the magnetic dip. 
    These are known as the elements of the Earth’s magnetic field.

  • Consider a material placed in an external magnetic field begin mathsize 12px style stack text B end text subscript text 0 end text end subscript with rightwards arrow on top end style
    The magnetic intensity is defined as

    begin mathsize 12px style straight H with rightwards arrow on top equals straight B with rightwards arrow on top subscript 0 over µ subscript 0 end style

    The magnetisation begin mathsize 12px style stack text M end text with rightwards arrow on top end style of the material is its dipole moment per unit volume. 
    The magnetic field begin mathsize 12px style stack text B end text with rightwards arrow on top end style in the material is

    begin mathsize 12px style straight B with rightwards arrow on top equals µ subscript 0 open parentheses straight H with rightwards arrow on top plus straight M with rightwards arrow on top close parentheses end style

    For a linear material begin mathsize 12px style stack text M end text with rightwards arrow on top equals straight chi straight H with rightwards arrow on top end style So,begin mathsize 12px style stack text B end text with rightwards arrow on top equals µ subscript straight r straight H with rightwards arrow on top end style
    where
    𝓍: Magnetic susceptibility of the material 
    μr: Relative magnetic permeability
    The relative magnetic permeability μr and the magnetic permeability μ are related as follows:

    μ = μ0 μr
    μr = 1 +𝓍

  • Magnetic materials are broadly classified as diamagnetic, paramagnetic and ferromagnetic. 
    For diamagnetic materials, 𝓍 is negative and small.
    For paramagnetic materials, 𝓍 is positive and small. 
    For ferromagnetic materials, 𝓍 is positive and large.

  • Substances which at room temperature retain their ferromagnetic property for a long period of time are called permanent magnets.

  • HYSTERESIS
    If a ferromagnetic material is magnetised in one direction and the applied magnetising field is removed, then its magnetisation will not be reduced to zero. It must be driven back to zero by a field in the opposite direction. If an alternating magnetic field intensity is applied to the material, its
    magnetisation will trace out a loop called a hysteresis loop. The phenomenon in which magnetic flux density (B) lags behind the magnetising field (H) in a ferromagnetic material during cycles of magnetisation is called hysteresis

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