Question

A long wire is bent into shape ABCDE as shown in fig., with
BCD being a semicircle with centre O and radius r metre. A
current of I amp. flows through it in the direction
A → B → C → D → E. Then the magnetic induction at the
point O of the figure in vacuum is
(a) μ₀ [I/2πr + 1/4r]
(b) μ₀ [I/2πr - 1/4r]
(c) μ₀ I / 4r
(d) μ₀ I / πr

Answer 1

Given that,

A long wire is bent into shape ABCDE. BCD being a semicircle with centre O.

Radius = r

Current = I

Magnetic field at O due to the semi-infinite segment AB is half of that due to an infinite wire

The magnetic induction is

$B_{AB}=\dfrac{1}{2}\times\dfrac{\mu_{0}i}{2\pi r}(-k)$

Magnetic field at O due to the semi-infinite segment DE is half of that due to an infinite wire

The magnetic induction is

$B_{DE}=\dfrac{1}{2}\times\dfrac{\mu_{0}i}{2\pi r}(-k)$

Magnetic field at O due to the semi-infinite segment BCD is half of that due to an infinite wire

The magnetic induction is

$B_{BCD}=\dfrac{1}{2}\times\dfrac{\mu_{0}i}{2r}$

The magnetic field due to semi circular portion is half of that due to a circular wire.

We need to calculate the magnetic induction at the  point O

Using formula for magnetic induction

$B_{total}=B_{AB}+B_{DE}+B_{BCD}$

Put the value into formula

$B_{total}=\dfrac{1}{2}\times\dfrac{\mu_{0}i}{2\pi r}(-k)+\dfrac{1}{2}\times\dfrac{\mu_{0}i}{2\pi r}(-k)+\dfrac{1}{2}\times\dfrac{\mu_{0}i}{2r}$

$B_{total}=\mu_{0}i(\dfrac{1}{2\pi r}+\dfrac{1}{4r})$

Hence, The magnetic induction at the  point O is $\mu_{0}i(\dfrac{1}{2\pi r}+\dfrac{1}{4r})$

(a) is correct option.

Answer 2

Answer:

Given that,

A long wire is bent into shape ABCDE. BCD being a semicircle with centre O.

Radius = r

Current = I

Magnetic field at O due to the semi-infinite segment AB is half of that due to an infinite wire

The magnetic induction is

Magnetic field at O due to the semi-infinite segment DE is half of that due to an infinite wire

The magnetic induction is

Magnetic field at O due to the semi-infinite segment BCD is half of that due to an infinite wire

The magnetic induction is

The magnetic field due to semi circular portion is half of that due to a circular wire.

We need to calculate the magnetic induction at the  point O

Using formula for magnetic induction

Put the value into formula

Hence, The magnetic induction at the  point O is  

(a) is correct option.