Biot-Savart Law

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JEE Advanced 2014

Comprehension Passage

The figure shows a circular loop of radius

a

with two long parallel wires (numbered 1 and 2) all in the plane of the paper. The distance of each wire from the centre of the loop is

d

. The loop and the wires are carrying the same current

I

. The current in the loop is in the counter-clockwise direction if seen from above.

Question 1:

When $d \approx a$ but wires are not touching the loop, it is found that the net magnetic field on the axis of the loop is zero at a height $h$ above the loop. In that case

(A)

currents in wire 1 and wire 2 are in the directions

P Q

and

R S

, respectively and

h \approx a

(B)

currents in wire 1 and wire 2 are in the directions

P Q

and

S R

, respectively and

h \approx a

(C)

currents in wire 1 and wire 2 are in the directions

P Q

and

S R

, respectively and

h \approx 1.2 a

(D)

currents in wire 1 and wire 2 are in the directions

P Q

and

R S

, respectively and

h \approx 1.2 a

Question 2:

Consider $d >> a$ , and the loop is rotated about its diameter parallel to the wires by $3 0^{\circ}$ from the position shown in the figure. If the currents in the wires are in the opposite directions, the torque on the loop at its new position will be (assume that the net field due to the wires is constant over the loop)

(A)

\frac{μ _{0} I ^{2} a ^{2}}{d}

(B)

\frac{μ _{0} I ^{2} a ^{2}}{2 d}

(C)

\frac{3 μ _{0} I ^{2} a ^{2}}{d}

(D)

\frac{3 μ _{0} I ^{2} a ^{2}}{2 d}

JEE Advanced 2018

LEVELJEE Advanced

Two infinitely long straight wires lie in the $x y$ -plane along the lines $x = \pm R$ . The wire located at $x = + R$ carries a constant current $I_{1}$ and the wire located at $x = - R$ carries a constant current $I_{2}$ . A circular loop of radius $R$ is suspended with its centre at $(0, 0, 3 R)$ and in a plane parallel to the $x y$ -plane. This loop carries a constant current $I$ in the clockwise direction as seen from above the loop. The current in the wire is taken to be positive, if it is in the $+ \hat{j}$ -direction. Which of the following statements regarding the magnetic field $B$ is (are) true?

* Multiple Correct Options

(A)

I_{1} = I_{2}

, then

B

cannot be equal to zero at the origin

(0, 0, 0)

(B)

I_{1} > 0

and

I_{2} < 0

, then

B

can be equal to zero at the origin

(0, 0, 0)

(C)

I_{1} < 0

and

I_{2} > 0

, then

B

can be equal to zero at the origin

(0, 0, 0)

(D)

I_{1} = I_{2}

, then the

z

-component of the magnetic field at the centre of the loop is

(- \frac{μ _{0} I}{2 R})

LEVELJEE Main

An infinitely long conductor $P QR$ is bent to form a right angle as shown in figure. A current $I$ flows through $P QR$ . The magnetic field due to this current at the point $M$ is $H_{1}$ . Now, another infinitely long straight conductor $QS$ is connected at $Q$ , so that current is $I /2$ in $QR$ as well as in $QS$ , the current in $P Q$ remaining unchanged. The magnetic field at $M$ is now $H_{2}$ . The ratio $H_{1} / H_{2}$ is given by

A steady current $I$ goes through a wire loop $P QR$ having shape of a right angle triangle with $P Q = 3 x$ , $P R = 4 x$ and $QR = 5 x$ . If the magnitude of the magnetic field at $P$ due to this loop is $k (\frac{μ _{0} I}{48 π x})$ , find the value of $k$ .

LEVELJEE Main

A battery is connected between two points $A$ and $B$ on the circumference of a uniform conducting ring of radius $r$ and resistance $R$ . One of the arcs $A B$ of the ring subtends an angle $θ$ at the centre. The value of the magnetic induction at the centre due to the current in the ring is

(A)

proportional to

(18 0^{\circ} - θ)

(B)

inversely proportional to

r

(C)

zero, only if

(θ = 18 0^{\circ})

(D)

zero for all values of

θ

JEE Main 2011

LEVELJEE Advanced

A long insulated copper wire is closely wound as a spiral of $N$ turns. The spiral has inner radius $a$ and outer radius $b$ . The spiral lies in the $X - Y$ plane and a steady current $I$ flows through the wire. The Z-component of the magnetic field at the centre of the spiral is

(A)

\frac{μ _{0} N I}{2 ( b - a )} ln (\frac{b}{a})

(B)

\frac{μ _{0} N I}{2 ( b - a )} ln (\frac{b + a}{b - a})

(C)

\frac{μ _{0} N I}{2 b} ln (\frac{b}{a})

(D)

\frac{μ _{0} N I}{2 b} ln (\frac{b + a}{b - a})

LEVELJEE Advanced

A non-planar loop of conducting wire carrying a current $I$ is placed as shown in the figure. Each of the straight sections of the loop is of length $2 a$ . The magnetic field due to this loop at the point $P (a, 0, a)$ points in the direction

(A)

\frac{1}{2} (- \hat{j} + \hat{k})

(B)

\frac{1}{3} (- \hat{j} + \hat{k} + \hat{i})

(C)

\frac{1}{3} (\hat{i} + \hat{j} + \hat{k})

(D)

\frac{1}{2} (\hat{i} + \hat{k})

JEE Advanced 2002

LEVELJEE Main

A long straight wire along the z-axis carries a current $I$ in the negative z-direction. The magnetic vector field $B$ at a point having coordinate $(x, y)$ on the $z = 0$ plane is

(A)

\frac{μ _{0} I ( y i ^ - x j ^ )}{2 π ( x ^{2} + y ^{2} )}

(B)

\frac{μ _{0} I ( x i ^ + y j ^ )}{2 π ( x ^{2} + y ^{2} )}

(C)

\frac{μ _{0} I ( x j ^ - y i ^ )}{2 π ( x ^{2} + y ^{2} )}

(D)

\frac{μ _{0} I ( x i ^ - y j ^ )}{2 π ( x ^{2} + y ^{2} )}

LEVELJEE Advanced

A pair of stationary and infinitely long bent wires are placed in the $x y$ plane as shown in figure. The wires carry current of $i = 10 A$ each as shown. The segments $L$ and $M$ are along the $x$ -axis. The segments $P$ and $Q$ are parallel to the $y$ -axis such that $O S = O R = 0.02 m$ . Find the magnitude and direction of the magnetic induction at the origin $O$ .

JEE Advanced 2001

LEVELJEE Advanced

A current of 10 A flows around a closed path in a circuit which is in the horizontal plane as shown in the figure. The circuit consists of eight alternating arcs of radii $r_{1} = 0.08$ m and $r_{2} = 0.12$ m. Each subtends the same angle at the centre. (a) Find the magnetic field produced by this circuit at the centre. (b) An infinitely long straight wire carrying a current of 10 A is passing through the centre of the above circuit vertically with the direction of the current being into the plane of the circuit. What is the force acting on the wire at the centre due to the current in the circuit? What is the force acting on the arc $A C$ and the straight segment $C D$ due to the current at the centre?

JEE Advanced 1988

LEVELJEE Main

The wire loop PQRSP formed by joining two semicircular wires of radii $R_{1}$ and $R_{2}$ carries a current $I$ as shown. The magnitude of the magnetic induction at the centre $C$ is ......

JEE Advanced 2007

LEVELJEE Main

Two wires each carrying a steady current $I$ are shown in four configurations in Column I. Some of the resulting effects are described in Column II. Match the statements in Column I with the statements in Column II.

List-I

(P)

Point

P

is situated midway between the wires.

(Q)

Point

P

is situated at the mid-point of the line joining the centres of the circular wires, which have same radii.

(R)

Point

P

is situated at the mid-point of the line joining the centres of the circular wires, which have same radii.

(S)

Point

P

is situated at the common centre of the wires.

List-II

(1)

The magnetic fields (

B

) at

P

due to the currents in the wires are in the same direction.

(2)

The magnetic fields (

B

) at

P

due to the currents in the wires are in opposite directions.

(3)

There is no magnetic field at

P

(4)

The wires repel each other.

JEE Advanced 1990

LEVELJEE Advanced

Two long parallel wires carrying currents 2.5 A and I (ampere) in the same direction (directed into the plane of the paper) are held at P and Q respectively such that they are perpendicular to the plane of paper. The points P and Q are located at a distance of 5 m and 2 m respectively from a collinear point R (see figure). (a) An electron moving with a velocity of $4 \times 1 0^{5}$ m/s along the positive x-direction experiences a force of magnitude $3.2 \times 1 0^{- 20}$ N at the point R. Find the value of I. (b) Find all the positions at which a third long parallel wire carrying a current of magnitude 2.5 A may be placed, so that the magnetic induction at R is zero.

LEVELJEE Main

A coil having $N$ turns is wound tightly in the form of a spiral with inner and outer radii $a$ and $b$ respectively. When a current $I$ passes through the coil, the magnetic field at the centre is

(A)

\frac{μ _{0} N I}{b}

(B)

\frac{2 μ _{0} N I}{a}

(C)

\frac{μ _{0} N I}{2 ( b - a )} ln (\frac{b}{a})

(D)

\frac{μ _{0} I}{2 ( b - a )} ln (\frac{b}{a})

JEE Advanced 2022

LEVELJEE Advanced

Which one of the following options represents the magnetic field $B$ at O due to the current flowing in the given wire segments lying on the xy plane?

(A)

\frac{- μ _{0} I}{L} (\frac{3}{2} + \frac{1}{4 2 π}) \hat{k}

(B)

\frac{- μ _{0} I}{L} (\frac{3}{2} + \frac{1}{2 2 π}) \hat{k}

(C)

\frac{- μ _{0} I}{L} (1 + \frac{1}{4 2 π}) \hat{k}

(D)

\frac{- μ _{0} I}{L} (1 + \frac{1}{4 π}) \hat{k}

JEE Advanced 2021

LEVELJEE Main

Two concentric circular loops, one of radius R and the other of radius 2R, lie in the xy-plane with the origin as their common center, as shown in the figure. The smaller loop carries current $I_{1}$ in the anti-clockwise direction and the larger loop carries current $I_{2}$ in the clockwise direction, with $I_{2} > 2 I_{1}$ . $B (x, y)$ denotes the magnetic field at a point $(x, y)$ in the xy-plane. Which of the following statement(s) is(are) current?

* Multiple Correct Options

(A)

B (x, y)

is perpendicular to the xy-plane at any point in the plane

(B)

∣ B (x, y) ∣

depends on x and y only through the radial distance

r = x^{2} + y^{2}

(C)

∣ B (x, y) ∣

is non-zero at all points for

r < R

(D)

B (x, y)

points normally outward from the xy-plane for all the points between the two loops

JEE Main 2007

LEVELJEE Main

Two identical conducting wires AOB and COD are placed at right angles to each other. The wire AOB carries an electric current $I_{1}$ and COD carries a current $I_{2}$ . The magnetic field on a point lying at a distance $d$ from O, in a direction perpendicular to the plane of the wires AOB and COD, will be given by

(A)

\frac{μ _{0}}{2 π} (\frac{I _{1} + I _{2}}{d})^{1/2}

(B)

\frac{μ _{0}}{2 π d} (I_{1}^{2} + I_{2}^{2})^{1/2}

(C)

\frac{μ _{0}}{2 π d} (I_{1} + I_{2})

(D)

\frac{μ _{0}}{2 π d} (I_{1}^{2} + I_{2}^{2})

JEE Main 2021

LEVELJEE Advanced

There are two infinitely long straight current carrying conductors and they are held at right angles to each other so that their common ends meet at the origin as shown in the figure given below. The ratio of current in both conductors is 1 : 1. The magnetic field at point P is

(A)

\frac{μ _{0} I}{4 π x y} [x^{2} + y^{2} + (x + y)]

(B)

\frac{μ _{0} I}{4 π x y} [x^{2} + y^{2} - (x + y)]

(C)

\frac{μ _{0} I x y}{4 π} [x^{2} + y^{2} - (x + y)]

(D)

\frac{μ _{0} I x y}{4 π} [x^{2} + y^{2} + (x + y)]

JEE Main 2021

LEVELJEE Main

A current of $1.5 A$ is flowing through a triangle, of side $9 cm$ each. The magnetic field at the centroid of the triangle is (Assume that, the current is flowing in the clockwise direction.)

(A)

3 \times 1 0^{- 7} T

, outside the plane of triangle

(B)

23 \times 1 0^{- 7} T

, outside the plane of triangle

(C)

23 \times 1 0^{- 5} T

, inside the plane of triangle

(D)

3 \times 1 0^{- 5} T

, inside the plane of triangle

LEVELJEE Main

Two concentric coils each of radius equal to $2 π$ cm are placed at right angles to each other. 3 A and 4 A are the currents flowing in each coil respectively. The magnetic induction in $Wb/m^{2}$ at the centre of the coils will be ( $μ_{0} = 4 π \times 1 0^{- 7} Wb/Am$ )

(A)

12 \times 1 0^{- 5}

(B)

1 0^{- 5}

(C)

5 \times 1 0^{- 5}

(D)

7 \times 1 0^{- 5}

JEE Main 2021

LEVELJEE Advanced

A coil having $N$ turns is wound tightly in the form of a spiral with inner and outer radii $a$ and $b$ , respectively. Find the magnetic field at centre, when a current $I$ passes through coil

(A)

\frac{μ _{0} I N}{2 ( b - a )} ln (\frac{b}{a})

(B)

\frac{μ _{0} I}{8} ln [\frac{a + b}{a - b}]

(C)

\frac{μ _{0} I}{4 ( a - b )} ln [\frac{1}{a} - \frac{1}{b}]

(D)

\frac{μ _{0} I}{8} ln (\frac{a - b}{a + b})

JEE Main 2019

LEVELJEE Main

The magnitude of the magnetic field at the centre of an equilateral triangular loop of side 1 m which is carrying a current of 10 A is [Take, $μ_{0} = 4 π \times 1 0^{- 7} NA^{- 2}$ ]

(A)

9 μ T

(B)

1 μ T

(C)

3 μ T

(D)

18 μ T

JEE Main 2020

LEVELJEE Advanced

A very long wire ABDMNDC is shown in figure carrying current $I$ . $A B$ and $B C$ parts are straight, long and at right angle. At $D$ wire forms a circular turn $D M N D$ of radius $R$ . $A B$ , $B C$ parts are tangential to circular turn at $N$ and $D$ . Magnetic field at the centre of circle is

(A)

\frac{μ _{0} I}{2 π R} (π + \frac{1}{2})

(B)

\frac{μ _{0} I}{2 R}

(C)

\frac{μ _{0} I}{2 π R} (π + 1)

(D)

\frac{μ _{0} I}{2 π R} (π - \frac{1}{2})

JEE Main 2019

LEVELJEE Main

As shown in the figure, two infinitely long, identical wires are bent by $9 0^{\circ}$ and placed in such a way that the segments $L P$ and $QM$ are along the X-axis, while segments $P S$ and $QN$ are parallel to the Y-axis. If $O P = O Q = 4 cm$ and the magnitude of the magnetic field at $O$ is $1 0^{- 4} T$ and the two wires carry equal currents (see figure), the magnitude of the current in each wire and the direction of the magnetic field at $O$ will be (Take, $μ_{0} = 4 π \times 1 0^{- 7} NA^{- 2}$ )

(A)

40 A, perpendicular out of the page

(B)

20 A, perpendicular into the page

(C)

20 A, perpendicular out of the page

(D)

40 A, perpendicular into the page

JEE Main 2020

LEVELJEE Main

Magnitude of magnetic field (in SI unit) at the centre of a hexagonal shaped coil of side 10 cm, 50 turns and carrying current $I$ ampere in units of $\frac{μ _{0} I}{π}$ is

(A)

2503

(B)

503

(C)

5003

(D)

53

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } When d≈a but wires are not touching the loop, it is found that the net magnetic field on the axis of the loop is zero at a height h above the loop. In that case

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A long straight wire along the z-axis carries a current I in the negative z-direction. The magnetic vector field B at a point having coordinate (x,y) on the z=0 plane is

List-I

List-II

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Which one of the following options represents the magnetic field B at O due to the current flowing in the given wire segments lying on the xy plane?

There are two infinitely long straight current carrying conductors and they are held at right angles to each other so that their common ends meet at the origin as shown in the figure given below. The ratio of current in both conductors is 1 : 1. The magnetic field at point P is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The magnitude of the magnetic field at the centre of an equilateral triangular loop of side 1 m which is carrying a current of 10 A is [Take, μ0​=4π×10−7 NA−2]

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Magnitude of magnetic field (in SI unit) at the centre of a hexagonal shaped coil of side 10 cm, 50 turns and carrying current I ampere in units of πμ0​I​ is

When $d \approx a$ but wires are not touching the loop, it is found that the net magnetic field on the axis of the loop is zero at a height $h$ above the loop. In that case

A long straight wire along the z-axis carries a current $I$ in the negative z-direction. The magnetic vector field $B$ at a point having coordinate $(x, y)$ on the $z = 0$ plane is

Which one of the following options represents the magnetic field $B$ at O due to the current flowing in the given wire segments lying on the xy plane?

The magnitude of the magnetic field at the centre of an equilateral triangular loop of side 1 m which is carrying a current of 10 A is [Take, $μ_{0} = 4 π \times 1 0^{- 7} NA^{- 2}$ ]

Magnitude of magnetic field (in SI unit) at the centre of a hexagonal shaped coil of side 10 cm, 50 turns and carrying current $I$ ampere in units of $\frac{μ _{0} I}{π}$ is