Physics-
General
Easy

Question

Consider a thin square sheet of side L and thickness t comma made of a material of resistivity rho. The resistance between two opposite faces, shown by the shaded areas in the figure is

  1. Directly proportional to L    
  2. Directly proportional to t    
  3. Independent of L    
  4. Independent of t    

The correct answer is: Independent of L


    R equals fraction numerator rho open parentheses L close parentheses over denominator A end fraction equals fraction numerator rho L over denominator t L end fraction equals fraction numerator rho over denominator t end fraction
    i e comma blank R is independent of L.
    Hence the correct option is (c).

    Related Questions to study

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    physics-

    The time taken by a block of wood (initially at rest)to slide down a smooth inclined plane 9.8 blank mlong (angle of inclination is 30 degree) is

    For one dimensional motion along a plane
    S equals u t plus fraction numerator 1 over denominator 2 end fraction a t to the power of 2 end exponent rightwards double arrow 9.8 equals 0 plus fraction numerator 1 over denominator 2 end fraction g sin invisible function application blank 30 degree t to the power of 2 end exponent rightwards double arrow t equals 2 s e c

    The time taken by a block of wood (initially at rest)to slide down a smooth inclined plane 9.8 blank mlong (angle of inclination is 30 degree) is

    physics-General
    For one dimensional motion along a plane
    S equals u t plus fraction numerator 1 over denominator 2 end fraction a t to the power of 2 end exponent rightwards double arrow 9.8 equals 0 plus fraction numerator 1 over denominator 2 end fraction g sin invisible function application blank 30 degree t to the power of 2 end exponent rightwards double arrow t equals 2 s e c
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    A particle starts from rest. Its acceleration left parenthesis a right parenthesis versus time left parenthesis t right parenthesis is as shown in the figure. The maximum speed of the particle will be

    The area under acceleration time graph gives change in velocity. As acceleration is zero at the end of 11 blank s e c

    i.e. v subscript m a x end subscript equalsArea ofblank increment O A B
    equals fraction numerator 1 over denominator 2 end fraction cross times 11 cross times 10 equals 55 blank m divided by s

    A particle starts from rest. Its acceleration left parenthesis a right parenthesis versus time left parenthesis t right parenthesis is as shown in the figure. The maximum speed of the particle will be

    Physics-General
    The area under acceleration time graph gives change in velocity. As acceleration is zero at the end of 11 blank s e c

    i.e. v subscript m a x end subscript equalsArea ofblank increment O A B
    equals fraction numerator 1 over denominator 2 end fraction cross times 11 cross times 10 equals 55 blank m divided by s
    General
    physics-

    v minus t graph for a particle is as shown. The distance travelled in the first 4 blank s is

    Distance covered equalsArea enclosed by v minus t graph
    equals Area of triangle equals fraction numerator 1 over denominator 2 end fraction cross times 4 cross times 8 equals 16 blank m

    v minus t graph for a particle is as shown. The distance travelled in the first 4 blank s is

    physics-General
    Distance covered equalsArea enclosed by v minus t graph
    equals Area of triangle equals fraction numerator 1 over denominator 2 end fraction cross times 4 cross times 8 equals 16 blank m
    parallel
    General
    Maths-

    Period of sin space x sin space open parentheses 120 to the power of ring operator minus x close parentheses sin space open parentheses 120 to the power of ring operator plus x close parentheses is

    sin space x sin space open parentheses 120 to the power of ring operator minus x close parentheses sin space open parentheses 120 to the power of ring operator plus x close parentheses
equals sin x space left parenthesis sin squared 120 degree minus sin squared x right parenthesis
equals sin x space open square brackets open parentheses fraction numerator square root of 3 over denominator 2 end fraction close parentheses squared minus sin squared x close square brackets
equals sin x space open square brackets 3 over 4 minus sin squared x close square brackets
equals 1 fourth open square brackets 3 space sin x space minus 4 sin squared x close square brackets
equals 1 fourth sin 3 x
f u n d a m e n t a l space p e r i o d space o f space sin space a x space i s space fraction numerator 2 straight pi over denominator a end fraction.
S o comma space p e r i o d space o f space 1 fourth sin 3 x space i s fraction numerator 2 straight pi over denominator 3 end fraction.

    Period of sin space x sin space open parentheses 120 to the power of ring operator minus x close parentheses sin space open parentheses 120 to the power of ring operator plus x close parentheses is

    Maths-General
    sin space x sin space open parentheses 120 to the power of ring operator minus x close parentheses sin space open parentheses 120 to the power of ring operator plus x close parentheses
equals sin x space left parenthesis sin squared 120 degree minus sin squared x right parenthesis
equals sin x space open square brackets open parentheses fraction numerator square root of 3 over denominator 2 end fraction close parentheses squared minus sin squared x close square brackets
equals sin x space open square brackets 3 over 4 minus sin squared x close square brackets
equals 1 fourth open square brackets 3 space sin x space minus 4 sin squared x close square brackets
equals 1 fourth sin 3 x
f u n d a m e n t a l space p e r i o d space o f space sin space a x space i s space fraction numerator 2 straight pi over denominator a end fraction.
S o comma space p e r i o d space o f space 1 fourth sin 3 x space i s fraction numerator 2 straight pi over denominator 3 end fraction.
    General
    physics-

    A particle starts from rest at t equals 0 and undergoes an acceleration a in m s to the power of negative 2 end exponent with time t in seconds which is as shown

    Which one of the following plot represents velocity V in m s to the power of negative 1 end exponent versus time t in seconds


    Takingthe motion from 0 to 2 blank s
    u equals 0 comma blank a equals 3 m s to the power of negative 2 end exponent comma blank t equals 2 s comma blank v equals ?
    v equals u plus a t equals 0 plus 3 cross times 2 equals 6 m s to the power of negative 1 end exponent
    Taking the motion from 2 blank s to 4 blank s
    v equals 6 plus open parentheses negative 3 close parentheses open parentheses 2 close parentheses equals 0 m s to the power of negative 1 end exponent

    A particle starts from rest at t equals 0 and undergoes an acceleration a in m s to the power of negative 2 end exponent with time t in seconds which is as shown

    Which one of the following plot represents velocity V in m s to the power of negative 1 end exponent versus time t in seconds

    physics-General

    Takingthe motion from 0 to 2 blank s
    u equals 0 comma blank a equals 3 m s to the power of negative 2 end exponent comma blank t equals 2 s comma blank v equals ?
    v equals u plus a t equals 0 plus 3 cross times 2 equals 6 m s to the power of negative 1 end exponent
    Taking the motion from 2 blank s to 4 blank s
    v equals 6 plus open parentheses negative 3 close parentheses open parentheses 2 close parentheses equals 0 m s to the power of negative 1 end exponent
    General
    Maths-

    Period of cos to the power of 6 space x plus sin to the power of 6 space x is

    By fundamental property of f(x) periodic function with period T.
    f (x+T) =f(x)
    let f (x) =cos to the power of 6 space x plus sin to the power of 6 space x
    assuming from the options smallest period of f(x) to be pi over 2.
    rightwards double arrow f left parenthesis x plus straight pi over 2 right parenthesis equals f left parenthesis x right parenthesis
rightwards double arrow cos to the power of 6 open parentheses x plus straight pi over 2 close parentheses plus sin to the power of 6 open parentheses x plus straight pi over 2 close parentheses equals cos to the power of 6 x plus sin to the power of 6 x
t h e r e f o r e space straight pi over 2 space i s space t h e space f u n d a m e n t a l space p e r i o d space o f space f left parenthesis x right parenthesis.

    Period of cos to the power of 6 space x plus sin to the power of 6 space x is

    Maths-General
    By fundamental property of f(x) periodic function with period T.
    f (x+T) =f(x)
    let f (x) =cos to the power of 6 space x plus sin to the power of 6 space x
    assuming from the options smallest period of f(x) to be pi over 2.
    rightwards double arrow f left parenthesis x plus straight pi over 2 right parenthesis equals f left parenthesis x right parenthesis
rightwards double arrow cos to the power of 6 open parentheses x plus straight pi over 2 close parentheses plus sin to the power of 6 open parentheses x plus straight pi over 2 close parentheses equals cos to the power of 6 x plus sin to the power of 6 x
t h e r e f o r e space straight pi over 2 space i s space t h e space f u n d a m e n t a l space p e r i o d space o f space f left parenthesis x right parenthesis.
    parallel
    General
    physics-

    Two inclined planes are located as shown in figure. A particle is projected from the foot of one frictionless plane along its line with a velocity just sufficient to carry it to top after which the particle slides down the other frictionless inclined plane. The total time it will take to reach the point C is

    The time of ascent = time of descent equals t subscript 0 end subscript
    T equals total time of flight equals 2 t subscript 0 end subscript

    sin invisible function application 45 degree equals fraction numerator 9.8 over denominator B C end fraction equals fraction numerator 9.8 over denominator s end fraction
    therefore blank s equals 9.8 square root of 2
    therefore blank s equals u t plus fraction numerator 1 over denominator 2 end fraction a t to the power of 2 end exponent
    s equals 0 cross times t plus fraction numerator 1 over denominator 2 end fraction left parenthesis g sin invisible function application 45 degree right parenthesis t subscript 0 end subscript superscript 2 end superscript
    or 9.8 square root of 2 equals fraction numerator 9.8 over denominator 2 square root of 2 end fraction t subscript 0 end subscript superscript 2 end superscript
    therefore blank t subscript 0 end subscript superscript 2 end superscript equals 4
    therefore blank t subscript 0 end subscript equals 2 s
    therefore blank T equals 2 t subscript 0 end subscript minus 4 s

    Two inclined planes are located as shown in figure. A particle is projected from the foot of one frictionless plane along its line with a velocity just sufficient to carry it to top after which the particle slides down the other frictionless inclined plane. The total time it will take to reach the point C is

    physics-General
    The time of ascent = time of descent equals t subscript 0 end subscript
    T equals total time of flight equals 2 t subscript 0 end subscript

    sin invisible function application 45 degree equals fraction numerator 9.8 over denominator B C end fraction equals fraction numerator 9.8 over denominator s end fraction
    therefore blank s equals 9.8 square root of 2
    therefore blank s equals u t plus fraction numerator 1 over denominator 2 end fraction a t to the power of 2 end exponent
    s equals 0 cross times t plus fraction numerator 1 over denominator 2 end fraction left parenthesis g sin invisible function application 45 degree right parenthesis t subscript 0 end subscript superscript 2 end superscript
    or 9.8 square root of 2 equals fraction numerator 9.8 over denominator 2 square root of 2 end fraction t subscript 0 end subscript superscript 2 end superscript
    therefore blank t subscript 0 end subscript superscript 2 end superscript equals 4
    therefore blank t subscript 0 end subscript equals 2 s
    therefore blank T equals 2 t subscript 0 end subscript minus 4 s
    General
    physics-

    I minus Vcharacteristic of a copper wire of length L and area of cross-section A is shown in figure. The slope of the curve becomes

    Slope of graph
    equals fraction numerator I over denominator V end fraction equals fraction numerator 1 over denominator R end fraction
    If experiment is performed at higher temperature then resistance increase and hence slope decrease, choice (a) is wrong.
    Similarly in choice (b) and (c) resistance increase.
    But for choice (d) resistance R increases, so slope decreases

    I minus Vcharacteristic of a copper wire of length L and area of cross-section A is shown in figure. The slope of the curve becomes

    physics-General
    Slope of graph
    equals fraction numerator I over denominator V end fraction equals fraction numerator 1 over denominator R end fraction
    If experiment is performed at higher temperature then resistance increase and hence slope decrease, choice (a) is wrong.
    Similarly in choice (b) and (c) resistance increase.
    But for choice (d) resistance R increases, so slope decreases
    General
    physics-

    Four concurrent coplanar forces in newton are acting at a point and keep it in equilibrium figure. Then values of F and theta are

    In equilibrium position along y-direction
    2 sin 60degree equals square root of 3 plus F cos invisible function application theta
    or 2 cross times fraction numerator square root of 3 over denominator 2 end fraction equals square root of 3 plus F cos invisible function application theta or F cos invisible function application theta equals 0
    As F not equal to 0
    therefore blank cos invisible function application theta equals 0 or theta equals 90 degree
    Along x-direction, F blank s i n 90 degree equals 1 plus 2 c o s 60 degree
    equals 1 plus 2 cross times fraction numerator 1 over denominator 2 end fraction
    F equals 2N

    Four concurrent coplanar forces in newton are acting at a point and keep it in equilibrium figure. Then values of F and theta are

    physics-General
    In equilibrium position along y-direction
    2 sin 60degree equals square root of 3 plus F cos invisible function application theta
    or 2 cross times fraction numerator square root of 3 over denominator 2 end fraction equals square root of 3 plus F cos invisible function application theta or F cos invisible function application theta equals 0
    As F not equal to 0
    therefore blank cos invisible function application theta equals 0 or theta equals 90 degree
    Along x-direction, F blank s i n 90 degree equals 1 plus 2 c o s 60 degree
    equals 1 plus 2 cross times fraction numerator 1 over denominator 2 end fraction
    F equals 2N
    parallel
    General
    physics-

    The effective capacitance between points X and Y shown in figure. Assuming C subscript 2 end subscript equals 10 blank muF and that outer capacitors are all 4 blank muF is

    The arrangement shows a Wheatstone bridge.
    As fraction numerator C subscript 1 end subscript over denominator C subscript 3 end subscript end fraction equals fraction numerator C subscript 4 end subscript over denominator C subscript 5 end subscript end fraction equals 1 comma blanktherefore the bridge is balanced.
    fraction numerator 1 over denominator C subscript s subscript 1 end subscript end subscript end fraction equals fraction numerator 1 over denominator 4 end fraction plus fraction numerator 1 over denominator 4 end fraction equals fraction numerator 2 over denominator 4 end fraction equals fraction numerator 1 over denominator 2 end fraction comma C subscript s subscript 1 end subscript end subscript equals 2 mu blank F
    Similarly, C subscript s end subscript subscript 2 end subscript equals 2 mu blank F
    therefore effective capacitance
    equals C subscript p end subscript equals C subscript s end subscript subscript 1 end subscript plus C subscript s end subscript subscript 2 end subscript equals 2 plus 2 plus equals 4 mu blank F

    The effective capacitance between points X and Y shown in figure. Assuming C subscript 2 end subscript equals 10 blank muF and that outer capacitors are all 4 blank muF is

    physics-General
    The arrangement shows a Wheatstone bridge.
    As fraction numerator C subscript 1 end subscript over denominator C subscript 3 end subscript end fraction equals fraction numerator C subscript 4 end subscript over denominator C subscript 5 end subscript end fraction equals 1 comma blanktherefore the bridge is balanced.
    fraction numerator 1 over denominator C subscript s subscript 1 end subscript end subscript end fraction equals fraction numerator 1 over denominator 4 end fraction plus fraction numerator 1 over denominator 4 end fraction equals fraction numerator 2 over denominator 4 end fraction equals fraction numerator 1 over denominator 2 end fraction comma C subscript s subscript 1 end subscript end subscript equals 2 mu blank F
    Similarly, C subscript s end subscript subscript 2 end subscript equals 2 mu blank F
    therefore effective capacitance
    equals C subscript p end subscript equals C subscript s end subscript subscript 1 end subscript plus C subscript s end subscript subscript 2 end subscript equals 2 plus 2 plus equals 4 mu blank F
    General
    physics-

    The four capacitors, each of 25 muF are connected as shown in figure. The DC voltmeter reads 200 V. the change on each plate of capacitor is

    Charge on each plate of each capacitor
    Q equals plus-or-minus C V equals plus-or-minus 25 cross times 10 to the power of negative 6 end exponent cross times 200
    equals plus-or-minus 5 cross times 10 to the power of negative 3 end exponent C

    The four capacitors, each of 25 muF are connected as shown in figure. The DC voltmeter reads 200 V. the change on each plate of capacitor is

    physics-General
    Charge on each plate of each capacitor
    Q equals plus-or-minus C V equals plus-or-minus 25 cross times 10 to the power of negative 6 end exponent cross times 200
    equals plus-or-minus 5 cross times 10 to the power of negative 3 end exponent C
    General
    physics-

    For the circuit shown in figure the charge on 4muF capacitor is

    Combined capacity of 1 blank mu F and 5 mu F blank= 1 + 5=6 blank mu F
    Now, 4 mu F blankand 6 mu F are in series.
    therefore blank fraction numerator 1 over denominator C subscript s end subscript end fraction equals fraction numerator 1 over denominator 4 end fraction plus fraction numerator 1 over denominator 6 end fraction plus fraction numerator 3 plus 2 over denominator 12 end fraction equals fraction numerator 5 over denominator 12 end fraction
    C subscript s end subscript equals fraction numerator 12 over denominator 5 end fraction mu F
    Charge in the arm containing 4 mu F blankcapacitor is
    q equals C subscript s end subscript cross times V equals fraction numerator 12 over denominator 5 end fraction cross times 10 equals 24 blank mu C

    For the circuit shown in figure the charge on 4muF capacitor is

    physics-General
    Combined capacity of 1 blank mu F and 5 mu F blank= 1 + 5=6 blank mu F
    Now, 4 mu F blankand 6 mu F are in series.
    therefore blank fraction numerator 1 over denominator C subscript s end subscript end fraction equals fraction numerator 1 over denominator 4 end fraction plus fraction numerator 1 over denominator 6 end fraction plus fraction numerator 3 plus 2 over denominator 12 end fraction equals fraction numerator 5 over denominator 12 end fraction
    C subscript s end subscript equals fraction numerator 12 over denominator 5 end fraction mu F
    Charge in the arm containing 4 mu F blankcapacitor is
    q equals C subscript s end subscript cross times V equals fraction numerator 12 over denominator 5 end fraction cross times 10 equals 24 blank mu C
    parallel
    General
    physics-

    The variation of electric potential with distance from a fixed point is shown in figure. What is the value of electric field at x=2 m.

    The variation of electric potential with distance from a fixed point is shown in figure. What is the value of electric field at x=2 m.

    physics-General
    General
    Maths-

    The integrating factor of the differential equation fraction numerator d y over denominator d x end fraction left parenthesis x log space x right parenthesis plus b equals 2 log space x is given by

    fraction numerator d y over denominator d x end fraction left parenthesis x log space x right parenthesis plus y equals 2 log space x
d i v i d i n g space x log x comma
rightwards double arrow fraction numerator d y over denominator d x end fraction plus fraction numerator y over denominator x space log x end fraction equals 2 over x
I F equals e to the power of integral fraction numerator 1 over denominator x space log x end fraction d x end exponent equals e to the power of log left parenthesis log x right parenthesis end exponent equals log space x

    The integrating factor of the differential equation fraction numerator d y over denominator d x end fraction left parenthesis x log space x right parenthesis plus b equals 2 log space x is given by

    Maths-General
    fraction numerator d y over denominator d x end fraction left parenthesis x log space x right parenthesis plus y equals 2 log space x
d i v i d i n g space x log x comma
rightwards double arrow fraction numerator d y over denominator d x end fraction plus fraction numerator y over denominator x space log x end fraction equals 2 over x
I F equals e to the power of integral fraction numerator 1 over denominator x space log x end fraction d x end exponent equals e to the power of log left parenthesis log x right parenthesis end exponent equals log space x
    General
    physics-

    As shown in figure, if the point C is earthed and the point A is given a potential of 2000 V, then the potential at point B will be

    Equivalent capacitance between points B a n d blank C is
    C to the power of ´ end exponent equals blank fraction numerator 10 cross times 10 over denominator 10 plus 10 end fraction plus 10 equals 15 mu F
    Now equivalent capacitance between points A a n d blank C is
    C to the power of ´ ´ end exponent equals fraction numerator 5 cross times 15 over denominator 15 plus 5 end fraction equals fraction numerator 75 over denominator 20 end fraction mu F
    Charge on capacitor of capacity 5mu F is
    Q equals C V equals fraction numerator 75 over denominator 20 end fraction cross times 2000 equals 7500 mu C
    (Since, potential at the point C will be zero)
    Now, potential difference across capacitor of 5mu F is
    V subscript A end subscript minus V subscript B end subscript blank equals fraction numerator Q over denominator 5 mu F end fraction equals fraction numerator 7500 mu C over denominator 5 mu C end fraction=1500volt
    As,V subscript A end subscript equals 2000volt
    Hence, V subscript B end subscript equals 2000 minus 1500 equals 500 volt.

    As shown in figure, if the point C is earthed and the point A is given a potential of 2000 V, then the potential at point B will be

    physics-General
    Equivalent capacitance between points B a n d blank C is
    C to the power of ´ end exponent equals blank fraction numerator 10 cross times 10 over denominator 10 plus 10 end fraction plus 10 equals 15 mu F
    Now equivalent capacitance between points A a n d blank C is
    C to the power of ´ ´ end exponent equals fraction numerator 5 cross times 15 over denominator 15 plus 5 end fraction equals fraction numerator 75 over denominator 20 end fraction mu F
    Charge on capacitor of capacity 5mu F is
    Q equals C V equals fraction numerator 75 over denominator 20 end fraction cross times 2000 equals 7500 mu C
    (Since, potential at the point C will be zero)
    Now, potential difference across capacitor of 5mu F is
    V subscript A end subscript minus V subscript B end subscript blank equals fraction numerator Q over denominator 5 mu F end fraction equals fraction numerator 7500 mu C over denominator 5 mu C end fraction=1500volt
    As,V subscript A end subscript equals 2000volt
    Hence, V subscript B end subscript equals 2000 minus 1500 equals 500 volt.
    parallel

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