Physics-
General
Easy

Question

A current i is flowing through the loop. The direction of the current and the shape of the loop are as shown in the figure.
The magnetic field at the centre of the loop is fraction numerator mu subscript 0 end subscript i over denominator R end fraction times.
left parenthesis M A equals R comma blank M B equals 2 R comma blank angle D M A equals 90 degree

  1. fraction numerator 5 over denominator 16 end fraction comma but out of the plane of the paper    
  2. fraction numerator 5 over denominator 16 end fraction comma but into the plane of the paper    
  3. fraction numerator 7 over denominator 16 end fraction comma but out of the plane of the paper    
  4. fraction numerator 7 over denominator 16 end fraction comma but into the plane of the paper    

The correct answer is: fraction numerator 7 over denominator 16 end fraction comma but into the plane of the paper


    (i) Magnetic field at the centre due to the curved portion D A equals fraction numerator mu subscript 0 end subscript i over denominator 4 pi R end fraction open parentheses fraction numerator 3 pi over denominator 2 end fraction close parentheses
    According to right hand screw rule, the magnetic field will be into the plane of paper.

    (ii) Magnetic field at M due to A B is zero.
    (iii) Magnetic field at the centre due to the curved portion B C is fraction numerator mu subscript 0 end subscript i over denominator 4 pi 2 R end fraction open parentheses fraction numerator pi over denominator 2 end fraction close parentheses. According to
    right hand screw rule, the magnetic field will be into the plane of paper.
    (iv) Magnetic field at M due to D C is zero.
    Hence, the resultant magnetic field at M
    equals fraction numerator 3 mu subscript 0 end subscript i over denominator 8 R end fraction plus 0 plus fraction numerator mu subscript 0 end subscript i over denominator 16 R end fraction plus 0 equals fraction numerator 7 mu subscript 0 end subscript i over denominator 16 R end fraction

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    B subscript 2 end subscript is also perpendicular to the page, directed upwards (right hand screw rule).
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    B subscript 1 end subscript equals fraction numerator mu subscript 0 end subscript i over denominator 2 pi r end fraction
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    A wire shown in figure carries a current of 40 A. If r equals 3.14 blank c m, the magnetic field at point P will be

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    B to the power of ´ ´ end exponent equals B subscript c end subscript cross times fraction numerator theta over denominator 2 pi end fraction equals fraction numerator mu subscript 0 end subscript over denominator 4 pi end fraction cross times fraction numerator 2 pi i over denominator e end fraction cross times fraction numerator theta over denominator 2 pi end fraction
    rightwards double arrow blank B to the power of ´ ´ end exponent equals blank fraction numerator mu subscript 0 end subscript over denominator 4 pi end fraction blank. fraction numerator i over denominator r end fraction theta
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    B to the power of ´ ´ end exponent equals B subscript c end subscript cross times fraction numerator theta over denominator 2 pi end fraction equals fraction numerator mu subscript 0 end subscript over denominator 4 pi end fraction cross times fraction numerator 2 pi i over denominator e end fraction cross times fraction numerator theta over denominator 2 pi end fraction
    rightwards double arrow blank B to the power of ´ ´ end exponent equals blank fraction numerator mu subscript 0 end subscript over denominator 4 pi end fraction blank. fraction numerator i over denominator r end fraction theta
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    theta equals 360 degree minus 90 degree equals 270 degree equals fraction numerator 3 pi over denominator 2 end fraction blank r a d.
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    If f colon R not stretchy rightwards arrow R is defined by f left parenthesis x right parenthesis equals fraction numerator x squared minus 4 over denominator x squared plus 1 end fraction text  then  end text f left parenthesis x right parenthesis text  is  end text

    f colon R not stretchy rightwards arrow R comma space f left parenthesis x right parenthesis equals fraction numerator x squared minus 4 over denominator x squared plus 1 end fraction
t h e n comma space f left parenthesis x subscript 1 right parenthesis equals f left parenthesis x subscript 2 right parenthesis rightwards double arrow fraction numerator x subscript 1 squared minus 4 over denominator x subscript 1 squared plus 1 end fraction equals fraction numerator x subscript 2 squared minus 4 over denominator x subscript 2 squared plus 1 end fraction
rightwards double arrow x subscript 1 squared x subscript 2 squared minus 4 x subscript 2 squared plus x subscript 1 squared minus 4 equals x subscript 1 squared x subscript 2 squared minus 4 x subscript 1 squared plus x subscript 2 squared minus 4
rightwards double arrow 5 x subscript 1 squared space minus 5 x subscript 2 squared equals 0
rightwards double arrow left parenthesis x subscript 1 minus x subscript 2 right parenthesis plus left parenthesis x subscript 1 plus x subscript 2 right parenthesis equals 0
rightwards double arrow x subscript 1 equals x subscript 2 space o r space x subscript 1 equals space minus x subscript 2
s o comma space f space i s space n o t space o n e minus o n e.

l e t space y element of R comma
a n d space y equals fraction numerator x squared minus 4 over denominator x squared plus 1 end fraction rightwards double arrow x squared y plus y equals x squared minus 4 rightwards double arrow 4 plus y equals x squared minus x squared y rightwards double arrow x squared equals fraction numerator 4 plus y over denominator 1 minus y end fraction rightwards double arrow x equals square root of fraction numerator 4 plus y over denominator 1 minus y end fraction end root
f open parentheses square root of fraction numerator 4 plus y over denominator 1 minus y end fraction end root close parentheses equals fraction numerator open parentheses square root of fraction numerator 4 plus y over denominator 1 minus y end fraction end root close parentheses squared minus 4 over denominator open parentheses square root of fraction numerator 4 plus y over denominator 1 minus y end fraction end root close parentheses squared plus 1 end fraction equals fraction numerator 4 plus y minus 4 plus 4 y over denominator 4 plus y plus 1 minus y end fraction equals fraction numerator 5 y over denominator 5 end fraction equals y
S o comma space f space i s space o n t o.

    If f colon R not stretchy rightwards arrow R is defined by f left parenthesis x right parenthesis equals fraction numerator x squared minus 4 over denominator x squared plus 1 end fraction text  then  end text f left parenthesis x right parenthesis text  is  end text

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    f colon R not stretchy rightwards arrow R comma space f left parenthesis x right parenthesis equals fraction numerator x squared minus 4 over denominator x squared plus 1 end fraction
t h e n comma space f left parenthesis x subscript 1 right parenthesis equals f left parenthesis x subscript 2 right parenthesis rightwards double arrow fraction numerator x subscript 1 squared minus 4 over denominator x subscript 1 squared plus 1 end fraction equals fraction numerator x subscript 2 squared minus 4 over denominator x subscript 2 squared plus 1 end fraction
rightwards double arrow x subscript 1 squared x subscript 2 squared minus 4 x subscript 2 squared plus x subscript 1 squared minus 4 equals x subscript 1 squared x subscript 2 squared minus 4 x subscript 1 squared plus x subscript 2 squared minus 4
rightwards double arrow 5 x subscript 1 squared space minus 5 x subscript 2 squared equals 0
rightwards double arrow left parenthesis x subscript 1 minus x subscript 2 right parenthesis plus left parenthesis x subscript 1 plus x subscript 2 right parenthesis equals 0
rightwards double arrow x subscript 1 equals x subscript 2 space o r space x subscript 1 equals space minus x subscript 2
s o comma space f space i s space n o t space o n e minus o n e.

l e t space y element of R comma
a n d space y equals fraction numerator x squared minus 4 over denominator x squared plus 1 end fraction rightwards double arrow x squared y plus y equals x squared minus 4 rightwards double arrow 4 plus y equals x squared minus x squared y rightwards double arrow x squared equals fraction numerator 4 plus y over denominator 1 minus y end fraction rightwards double arrow x equals square root of fraction numerator 4 plus y over denominator 1 minus y end fraction end root
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S o comma space f space i s space o n t o.
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    A copper wire of negligible mass, 1 m length and cross-sectional area 10 to the power of negative 6 end exponent is kept on a smooth horizontal table with one end fixed. A ball of mass 1 kg is attached to the other end. The wire and the ball are rotated with an angular velocity 20 rads to the power of negative 1 end exponent. blankIf the elongation in the wire is 10 to the power of negative 3 end exponent m, then the Young’s modulus is

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    Y equals fraction numerator F l over denominator A increment l end fraction equals blank fraction numerator open parentheses m l omega to the power of 2 end exponent close parentheses l over denominator A blank increment l end fraction o r blank Y equals fraction numerator m l to the power of 2 end exponent omega to the power of 2 end exponent over denominator A increment l end fraction
    Or Y equals blank fraction numerator 1 cross times 1 cross times 1 cross times 20 cross times 20 over denominator 10 to the power of negative 6 end exponent cross times 10 to the power of negative 3 end exponent end fraction equals 4 blank cross times 10 to the power of 11 end exponent N m to the power of negative 2 end exponent
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    Two wires of equal cross-section but one made of steel and the other of copper are joined end to end. When the combination is kept under tension, the elongations in the two wires are found to be equal. What is the ratio of the lengths of the two wires? (Given : steel = 2 cross times 10 to the power of 11 end exponent N m to the power of negative 2 end exponent right parenthesis

    Y equals fraction numerator s t r e s s over denominator s t r a i n end fraction blank o r blank s t r a i n equals fraction numerator s t r e s s over denominator Y end fraction o r fraction numerator increment L over denominator L end fraction equals fraction numerator s t r e s s over denominator Y end fraction
    Since, cross-sections are equal and same tension exists in both the wires, therefore, the stresses developed are equal.
    Also, increment L is given to be the same for both the wires.
    therefore blank L blank proportional to Y
    therefore blank fraction numerator L subscript s end subscript over denominator L subscript C u end subscript end fraction equals blank fraction numerator 2 blank cross times 10 to the power of 11 end exponent over denominator 1.1 blank cross times 10 to the power of 11 end exponent end fraction equals fraction numerator 20 over denominator 11 end fraction

    Two wires of equal cross-section but one made of steel and the other of copper are joined end to end. When the combination is kept under tension, the elongations in the two wires are found to be equal. What is the ratio of the lengths of the two wires? (Given : steel = 2 cross times 10 to the power of 11 end exponent N m to the power of negative 2 end exponent right parenthesis

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    Y equals fraction numerator s t r e s s over denominator s t r a i n end fraction blank o r blank s t r a i n equals fraction numerator s t r e s s over denominator Y end fraction o r fraction numerator increment L over denominator L end fraction equals fraction numerator s t r e s s over denominator Y end fraction
    Since, cross-sections are equal and same tension exists in both the wires, therefore, the stresses developed are equal.
    Also, increment L is given to be the same for both the wires.
    therefore blank L blank proportional to Y
    therefore blank fraction numerator L subscript s end subscript over denominator L subscript C u end subscript end fraction equals blank fraction numerator 2 blank cross times 10 to the power of 11 end exponent over denominator 1.1 blank cross times 10 to the power of 11 end exponent end fraction equals fraction numerator 20 over denominator 11 end fraction
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    The relation between gamma comma eta and K for a elastic material is

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    If f colon R not stretchy rightwards arrow R is defined by f left parenthesis x right parenthesis equals fraction numerator x squared minus 4 over denominator x squared plus 1 end fraction then f left parenthesis x right parenthesis text  is  end text

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    Two long parallel conductors carry currents in opposite directions as shown. One conductor carries a current of 10 A and the distance between the wires is d=10cm  Current I is adjusted, so that the magnetic field at P is zero. P is at a distance of 5 cm to the right of the 10 A current. Value of I is

    From Biot-Savart’s law the magnetic field open parentheses B close parentheses due to a conductor carrying current I comma at a distance r subscript 1 end subscript is
    B subscript 1 end subscript equals fraction numerator mu subscript 0 end subscript I subscript 1 end subscript over denominator 2 pi r subscript 1 end subscript end fraction
    Magnetic field at P due to current in second conductor is
    B subscript 2 end subscript equals fraction numerator mu subscript 0 end subscript I subscript 2 end subscript over denominator 2 pi open parentheses r subscript 1 end subscript plus d close parentheses end fraction
    From Fleming’s right hands rule the fields at P are directed opposite.
    therefore R e s u l t a n t s comma blank f i e l d blank B subscript 1 end subscript equals B subscript 2 end subscript
    therefore blank fraction numerator mu subscript 0 end subscript I subscript 1 end subscript over denominator 2 pi r subscript 1 end subscript end fraction equals blank fraction numerator mu subscript 0 end subscript I subscript 2 end subscript over denominator 2 pi open parentheses r subscript 1 end subscript plus d close parentheses end fraction
    Given, I subscript 1 end subscript equals 10 blank A comma blank r subscript 1 end subscript equals 5 comma blank r subscript 1 end subscript plus d equals 5 plus 10 equals 15 blank c m
    therefore blank I subscript 2 end subscript equals fraction numerator I subscript 1 end subscript over denominator r subscript 1 end subscript end fraction blank cross times open parentheses r subscript 1 end subscript plus d close parentheses
    I subscript 2 end subscript equals fraction numerator 10 over denominator 5 end fraction cross times 15 equals 30 blank A

    Two long parallel conductors carry currents in opposite directions as shown. One conductor carries a current of 10 A and the distance between the wires is d=10cm  Current I is adjusted, so that the magnetic field at P is zero. P is at a distance of 5 cm to the right of the 10 A current. Value of I is

    physics-General
    From Biot-Savart’s law the magnetic field open parentheses B close parentheses due to a conductor carrying current I comma at a distance r subscript 1 end subscript is
    B subscript 1 end subscript equals fraction numerator mu subscript 0 end subscript I subscript 1 end subscript over denominator 2 pi r subscript 1 end subscript end fraction
    Magnetic field at P due to current in second conductor is
    B subscript 2 end subscript equals fraction numerator mu subscript 0 end subscript I subscript 2 end subscript over denominator 2 pi open parentheses r subscript 1 end subscript plus d close parentheses end fraction
    From Fleming’s right hands rule the fields at P are directed opposite.
    therefore R e s u l t a n t s comma blank f i e l d blank B subscript 1 end subscript equals B subscript 2 end subscript
    therefore blank fraction numerator mu subscript 0 end subscript I subscript 1 end subscript over denominator 2 pi r subscript 1 end subscript end fraction equals blank fraction numerator mu subscript 0 end subscript I subscript 2 end subscript over denominator 2 pi open parentheses r subscript 1 end subscript plus d close parentheses end fraction
    Given, I subscript 1 end subscript equals 10 blank A comma blank r subscript 1 end subscript equals 5 comma blank r subscript 1 end subscript plus d equals 5 plus 10 equals 15 blank c m
    therefore blank I subscript 2 end subscript equals fraction numerator I subscript 1 end subscript over denominator r subscript 1 end subscript end fraction blank cross times open parentheses r subscript 1 end subscript plus d close parentheses
    I subscript 2 end subscript equals fraction numerator 10 over denominator 5 end fraction cross times 15 equals 30 blank A
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    Two long straight wires are set parallel to each other. Each carries a current i in the opposite direction and the separation between them is 2R. The intensity of the magnetic field midway between them is

    Magnetic field at mid-point due to wire A B
    B subscript 1 end subscript equals fraction numerator mu subscript 0 end subscript i over denominator 2 pi R end fraction
    Magnetic field at mid-point due to wire C D
    B subscript 2 end subscript equals fraction numerator mu subscript 0 end subscript i over denominator 2 pi R end fraction
    Resultant of Magnetic field B equals B subscript 1 end subscript plus B subscript 2 end subscript
    equals fraction numerator mu subscript 0 end subscript i over denominator 2 pi R end fraction plus fraction numerator mu subscript 0 end subscript i over denominator 2 pi R end fraction
    B equals fraction numerator mu subscript 0 end subscript i over denominator pi R end fraction

    Two long straight wires are set parallel to each other. Each carries a current i in the opposite direction and the separation between them is 2R. The intensity of the magnetic field midway between them is

    physics-General
    Magnetic field at mid-point due to wire A B
    B subscript 1 end subscript equals fraction numerator mu subscript 0 end subscript i over denominator 2 pi R end fraction
    Magnetic field at mid-point due to wire C D
    B subscript 2 end subscript equals fraction numerator mu subscript 0 end subscript i over denominator 2 pi R end fraction
    Resultant of Magnetic field B equals B subscript 1 end subscript plus B subscript 2 end subscript
    equals fraction numerator mu subscript 0 end subscript i over denominator 2 pi R end fraction plus fraction numerator mu subscript 0 end subscript i over denominator 2 pi R end fraction
    B equals fraction numerator mu subscript 0 end subscript i over denominator pi R end fraction
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    A current of 10 A is passing through a long wire which has semicircular loop of the radius 20 cm as shown in the figure. Magnetic field produced at the centre of the loop is

    B equals fraction numerator 40 over denominator 4 pi end fraction cross times fraction numerator pi i over denominator r end fraction equals 10 to the power of negative 7 end exponent cross times fraction numerator pi cross times 10 over denominator 20 cross times 10 to the power of negative 2 end exponent end fraction blank 5 pi mu t e s l a

    A current of 10 A is passing through a long wire which has semicircular loop of the radius 20 cm as shown in the figure. Magnetic field produced at the centre of the loop is

    physics-General
    B equals fraction numerator 40 over denominator 4 pi end fraction cross times fraction numerator pi i over denominator r end fraction equals 10 to the power of negative 7 end exponent cross times fraction numerator pi cross times 10 over denominator 20 cross times 10 to the power of negative 2 end exponent end fraction blank 5 pi mu t e s l a
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    In the figure shown, the magnetic field induction at the point O will be

    Field due to a straight wire of infinite length is fraction numerator mu subscript 0 end subscript i over denominator 4 pi r end fraction if the point is on a line perpendicular to its length while at the centre of a semicircular coil is fraction numerator mu subscript 0 end subscript pi i over denominator 4 pi r end fraction.

    therefore B equals B subscript a end subscript plus B subscript b end subscript plus B subscript c end subscript
    equals fraction numerator mu subscript 0 end subscript over denominator 4 pi end fraction fraction numerator i over denominator r end fraction plus blank fraction numerator mu subscript 0 end subscript over denominator 4 pi end fraction fraction numerator pi i over denominator r end fraction plus fraction numerator mu subscript 0 end subscript over denominator 4 pi end fraction fraction numerator i over denominator r end fraction
    equals fraction numerator mu subscript 0 end subscript over denominator 4 pi end fraction fraction numerator i over denominator r end fraction open parentheses pi plus 2 close parentheses out of the page

    In the figure shown, the magnetic field induction at the point O will be

    physics-General
    Field due to a straight wire of infinite length is fraction numerator mu subscript 0 end subscript i over denominator 4 pi r end fraction if the point is on a line perpendicular to its length while at the centre of a semicircular coil is fraction numerator mu subscript 0 end subscript pi i over denominator 4 pi r end fraction.

    therefore B equals B subscript a end subscript plus B subscript b end subscript plus B subscript c end subscript
    equals fraction numerator mu subscript 0 end subscript over denominator 4 pi end fraction fraction numerator i over denominator r end fraction plus blank fraction numerator mu subscript 0 end subscript over denominator 4 pi end fraction fraction numerator pi i over denominator r end fraction plus fraction numerator mu subscript 0 end subscript over denominator 4 pi end fraction fraction numerator i over denominator r end fraction
    equals fraction numerator mu subscript 0 end subscript over denominator 4 pi end fraction fraction numerator i over denominator r end fraction open parentheses pi plus 2 close parentheses out of the page
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    PQ and RS are long parallel conductors separated by certain distance. M is the mid-point between them (see the figure). The net magnetic field at M is B. Now, the current 2A is switched off. The field at M now becomes

    Magnetic field at mid-point M in first case is B equals B subscript P Q end subscript minus B subscript R S end subscript
    left parenthesis therefore B subscript P Q end subscript a n d blank B subscript R S end subscript a r e blank i n blank o p p o s i t e blank d i r e c t i o n s right parenthesis
    equals fraction numerator 4 blank mu subscript 0 end subscript over denominator 4 pi d end fraction minus fraction numerator 2 blank mu subscript 0 end subscript over denominator 4 pi d end fraction equals fraction numerator 2 blank mu subscript 0 end subscript over denominator 4 pi d end fraction
    When the current 2 A is switched off, the net magnetic field at M is due to current 1 A
    B to the power of ´ end exponent equals fraction numerator mu subscript 0 end subscript cross times 2 cross times 1 over denominator 4 pi d end fraction equals B

    PQ and RS are long parallel conductors separated by certain distance. M is the mid-point between them (see the figure). The net magnetic field at M is B. Now, the current 2A is switched off. The field at M now becomes

    physics-General
    Magnetic field at mid-point M in first case is B equals B subscript P Q end subscript minus B subscript R S end subscript
    left parenthesis therefore B subscript P Q end subscript a n d blank B subscript R S end subscript a r e blank i n blank o p p o s i t e blank d i r e c t i o n s right parenthesis
    equals fraction numerator 4 blank mu subscript 0 end subscript over denominator 4 pi d end fraction minus fraction numerator 2 blank mu subscript 0 end subscript over denominator 4 pi d end fraction equals fraction numerator 2 blank mu subscript 0 end subscript over denominator 4 pi d end fraction
    When the current 2 A is switched off, the net magnetic field at M is due to current 1 A
    B to the power of ´ end exponent equals fraction numerator mu subscript 0 end subscript cross times 2 cross times 1 over denominator 4 pi d end fraction equals B