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A current I flows in a long straight wir...

A current `I` flows in a long straight wire (into the plane of the paper) with cross-section having the from of a circular are of radius `R` a shown in the figure. Find the magnetic induction at the point `O`.

Text Solution

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We have the result of magnetic filed for a straight conductor. For a distribution, we have to make a differential equation and integrate.
Consider a small element of width `R d theta` as shown in the figure.
`dB=(mu_(0)(dI))/(2piR)`
`dI=(I)/((piR)/(2))d theta=(2I)/(pi)d theta :. dB=(mu_(0)Id theta)/(pi^(2)R)`
As we have to integrate `x` and `y` components separately.
`dB_(x)=dB costheta=(mu_(0)I)/(pi^(2)R)costheta d theta`
`dB_(y)=dB sin theta=(mu_(0)I)/(pi^(2)R)sintheta d theta`
`B_(x)=intdB_(x)=(mu_(0)I)/(pi^(2)R)int_(-pi//4)^(+pi//4)costheta d theta=(mu_(0)Isqrt(2))/(pi^(2)R)`
`B_(y)=intdB_(y)=(mu_(0)I)/(pi^(2)R)int_(-pi//4)^(+pi//4)sintheta d theta=0`
`vecB=(mu_(0)Isqrt(2))/(pi^(2)R)hati`

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Knowledge Check

  • A current I flows an infinitely long wire with cross section in the form of a semi - circular ring of radius R . The magnitude of the magnetic induction along its axis is :

    A
    `(mu_(0)I)/( 2 pi^(2) R )`
    B
    `(mu_(0)I)/( 2 pi R )`
    C
    `(mu_(0)I)/( 4 pi R )`
    D
    `(mu_(0)I)/( pi^(2) R )`

  • A current I flows through a long straight conductor which is bent into a circular loop of radius R in the middle as shown in the figure . The magnitude of the net magnetic field at point O will be

    A
    Zero
    B
    `(mu_0I)/(2R)(1+pi)`
    C
    `(mu_0I)/(4piR)`
    D
    `(mu_0I)/(2R)(1-1/pi)`

  • A current I, flowing in a long and straight wire along z-axis, is directed into two identical semicircular portions lying in y-z and x-z planes as shown in the figure. The magnetic field at the centre O is

    A
    zero
    B
    `(mu_0l)/(4pi)`
    C
    `(mu_0l)/(2sqrt2r)`
    D
    `(mu_0l)/(sqrt2r)`