JEE Advanced PYQ Physics – Magnetic Field Lines, Biot-Savart & Ampere's Law

Q 1 :

An infinitely long wire, located on the $z$ -axis, carries a current $I$ along the $+ z$ -direction and produces the magnetic field $\vec{B}$ . The magnitude of the line integral $\int \vec{B} \cdot d \vec{l}$ along a straight line from the point $(\sqrt{3} a, a, 0)$ to $(a, a, 0)$ is given by [ $μ_{0}$ is the magnetic permeability of free space.] [2024]

$\frac{7 μ_{0} I}{24}$
$\frac{7 μ_{0} I}{12}$
$\frac{μ_{0} I}{8}$
$\frac{μ_{0} I}{6}$

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(1)

From figure,

$| \vec{d ℓ} |$ $= r d θ$

Magnetic field, $| \vec{B} |$ $= \frac{μ_{0} I}{2 π r}$

$\int \vec{B} \cdot d \vec{ℓ} = \int | \vec{B} | | \vec{d ℓ} | \cos θ ° = \int (\frac{μ_{0} I}{2 π r}) (r d θ)$

$= \int_{θ_{1}}^{θ_{2}} \frac{μ_{0} I}{2 π} d θ = \frac{μ_{0} I}{2 π} [θ_{2} - (- θ_{1})]$ $[θ_{1} is anticlockwise hence taken negative]$

or, $\int \vec{B} \cdot d \vec{ℓ} = \frac{μ_{0} I}{2 π} (θ_{2} + θ_{1})$

Here, $\tan θ_{1} = \frac{a \sqrt{3}}{a} = \sqrt{3}$ $∴$ $θ_{1} = \frac{π}{3}$

and $\tan θ_{2} = \frac{a}{a} = 1$ $∴$ $θ_{2} = \frac{π}{4}$

Putting the values of $θ_{1}$ and $θ_{2}$ in Eq. (i),

$\int \vec{B} \cdot d \vec{ℓ} = \frac{μ_{0} I}{2 π} [\frac{π}{3} + \frac{π}{4}]$

or, $\int \vec{B} \cdot \vec{d ℓ} = \frac{μ_{0} I}{2 π} [\frac{4 π + 3 π}{12}] = \frac{7 μ_{0} I}{24}$

Q 2 :

Which one of the following options represents the magnetic field $\vec{B}$ at O due to the current flowing in the given wire segments lying on the $x y$ plane? [2022]

$\vec{B} = \frac{- μ_{0} I}{L} (\frac{3}{2} + \frac{1}{\sqrt[4]{2 π}}) \hat{k}$
$\vec{B} = \frac{- μ_{0} I}{L} (\frac{3}{2} + \frac{1}{2 \sqrt{2 π}}) \hat{k}$
$\vec{B} = \frac{- μ_{0} I}{L} (1 + \frac{1}{4 \sqrt{2 π}}) \hat{k}$
$\vec{B} = \frac{- μ_{0} I}{L} (1 + \frac{1}{4 π}) \hat{k}$

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(3)

$\vec{B} = \frac{μ_{0} I}{4 π L} \sin 45 ° (- \hat{k}) + \frac{μ_{0} I π}{4 π (\frac{L}{2})} (- \hat{k}) + \frac{μ_{0} I}{4 π (\frac{L}{4})} \times \frac{π}{2} (- \hat{k})$

$= \frac{- μ_{0} I}{L} [\frac{1}{4 π \sqrt{2}} + 1] \hat{k}$

Q 3 :

A symmetric star shaped conducting wire loop is carrying a steady state current $I$ as shown in the figure. The distance between the diametrically opposite vertices of the star is $4 a$ . The magnitude of the magnetic field at the center of the loop is [2017]

$\frac{μ_{0} I}{4 π a} 6 [\sqrt{3} - 1]$
$\frac{μ_{0} I}{4 π a} 6 [\sqrt{3} + 1]$
$\frac{μ_{0} I}{4 π a} 3 [\sqrt{3} - 1]$
$\frac{μ_{0} I}{4 π a} 3 [2 - \sqrt{3}]$

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(1)

From figure,

In $∆ O A C$ ,

$\cos 60 ° = \frac{O C}{O A} = \frac{O C}{2 a}$

$∴$ $O C = 2 a \times \frac{1}{2} = a$

Magnetic field at $' O'$ due to element AB

$= \frac{μ_{0}}{4 π} \frac{I}{a} (\sin 60 ° - \sin 30 °)$

$= \frac{μ_{0}}{4 π} \frac{I}{a} (\frac{\sqrt{3}}{2} - \frac{1}{2})$ $= \frac{μ_{0} I}{4 π a} \times \frac{1}{2} (\sqrt{3} - 1)$

$∴$ Magnetic field at the centre, due to the complete loop

$= [\frac{μ_{0} I}{4 π a} \times \frac{1}{2} (\sqrt{3} - 1)] \times 12$ $= \frac{μ_{0} I}{4 π a} 6 (\sqrt{3} - 1)$

Q 4 :

An infinitely long hollow conducting cylinder with inner radius $\frac{R}{2}$ and outer radius $R$ carries a uniform current density along its length. The magnitude of the magnetic field, $| \vec{B} |$ as a function of the radial distance $r$ from the axis is best represented by [2012]

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(4)

For $r < \frac{R}{2},$ $B = 0$

For $\frac{R}{2} \leq r < R,$

$B = \frac{μ_{0}}{2} [r - \frac{R^{2}}{2 r}] J$

For $r > R,$ $B = \frac{μ_{0} i}{2 π r}$

i.e., $B \propto \frac{1}{r}$

Hence graph (4) correctly depicts the $| \vec{B} |$ versus $r$ graph.

Q 5 :

A loop carrying current $I$ lies in the $x - y$ plane as shown in the figure. The unit vector $\hat{k}$ is coming out of the plane of the paper. The magnetic moment of the current loop is [2012]

$a^{2} I \hat{k}$
$(\frac{π}{2} + 1) a^{2} I \hat{k}$
$- (\frac{π}{2} + 1) a^{2} I \hat{k}$
$(2 π + 1) a^{2} I \hat{k}$

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(2)

Magnetic moment of a current-carrying loop $\vec{M} = N I \vec{A}$

Here, $N = 1,$ $A = a^{2} + 2 π {(\frac{a}{2})}^{2} = a^{2} (1 + \frac{π}{2})$

From the screw law, the direction of $m$ is outward or in the positive $z$ -direction.

$∴$ $\vec{M} = I a^{2} (1 + \frac{π}{2}) \hat{k}$

Q 6 :

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 XY plane and a steady current $' I'$ flows through the wire. The Z-component of the magnetic field at the centre of the spiral is [2011]

$\frac{μ_{0} N I}{2 (b - a)} \ln (\frac{b}{a})$
$\frac{μ_{0} N I}{2 (b - a)} \ln (\frac{b + a}{b - a})$
$\frac{μ_{0} N I}{2 b} \ln (\frac{b}{a})$
$\frac{μ_{0} N I}{2 b} \ln (\frac{b + a}{b - a})$

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(1)

Q 7 :

A long straight wire along the Z-axis carries a current $I$ in the negative Z-direction. The magnetic vector field $\vec{B}$ at a point having coordinates $(x, y)$ in the $Z = 0$ plane is [2002]

$\frac{μ_{0} I}{2 π} (\frac{y \hat{i} - x \hat{j}}{x^{2} + y^{2}})$
$\frac{μ_{0} I}{2 π} (\frac{x \hat{i} + y \hat{j}}{x^{2} + y^{2}})$
$\frac{μ_{0} I}{2 π} (\frac{x \hat{j} - y \hat{i}}{x^{2} + y^{2}})$
$\frac{μ_{0} I}{2 π} (\frac{x \hat{i} - y \hat{j}}{x^{2} + y^{2}})$

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(1)

The wire carries a current $I$ in the negative $z$ -direction. We have to consider the magnetic vector field $\vec{B}$ at $(x, y)$ in the $z = 0$ plane.

Magnetic field $\vec{B}$ is perpendicular to $O P$ .

$∴$ $\vec{B} = B \sin θ \hat{i} - B \cos θ \hat{j}$

$\sin θ = \frac{y}{r}, \cos θ = \frac{x}{r}, B = \frac{μ_{0} I}{2 π r}, x^{2} + y^{2} = r^{2}$

$∴$ $\vec{B} = \frac{μ_{0} I}{2 π r^{2}} (y \hat{i} - x \hat{j})$

or $\vec{B} = \frac{μ_{0} I (y \hat{i} - x \hat{j})}{2 π (x^{2} + y^{2})}$

Q 8 :

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 center is [2001]

$\frac{μ_{0} N I}{b}$
$\frac{2 μ_{0} N I}{a}$
$\frac{2 μ_{0} N I}{2 (b - a)} \ln (\frac{b}{a})$
$\frac{2 μ_{0} N I}{2 (b - a)} \ln (\frac{a}{b})$

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(3)

Let us consider an element of thickness $d x$ of wire. Let it be at a distance $x$ from the centre $O$ .

$Number of turns per unit length = \frac{N}{b - a}$

$∴$ $Number of turns in thickness d x = \frac{N}{b - a} d x$

Magnetic field due to this small element at $O$ ,

$d B = \frac{μ_{0}}{2} \cdot \frac{N I}{(b - a)} \cdot \frac{d x}{x}$

$B = \int_{a}^{b} \frac{μ_{0}}{2} \cdot \frac{N I}{(b - a)} \cdot \frac{d x}{x} = \frac{μ_{0}}{2} \cdot \frac{N I}{(b - a)} \int_{a}^{b} \frac{d x}{x}$

$= \frac{μ_{0}}{2} \cdot \frac{N I}{(b - a)} {[\log_{e} x]}_{a}^{b}$

$∴$ $B = \frac{μ_{0}}{2} \frac{N I}{(b - a)} \ln (\frac{b}{a})$

Q 9 :

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 [2001]

$\frac{1}{\sqrt{2}} (- \hat{j} + \hat{k})$
$\frac{1}{\sqrt{3}} (- \hat{j} + \hat{k} + \hat{i})$
$\frac{1}{\sqrt{3}} (\hat{i} + \hat{j} + \hat{k})$
$\frac{1}{\sqrt{2}} (\hat{i} + \hat{k})$

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(4)

Here loop $A D E F A$ lies in the $y - z$ plane and loop $A B C D A$ lies in the $x - y$ plane.

By choosing the loops, we find that in one loop we have to take current from $A$ to $D$ and in the other one from $D$ to $A$ . Effectively there is no current in $A D$ . Hence these two currents cancel out the effect of each other as far as creating magnetic field at the point $P$ is considered.

The point $(a, 0, a)$ lies in the $X - Z$ plane.

The magnetic field due to current in $A B C D A$ will be in the positive $Z$ -direction.

Due to symmetry, the $y$ -components and $x$ -components will cancel out each other.

Similarly, the magnetic field due to current in $A D E F A$ will be in the $x$ -direction.

$∴$ The direction of resultant magnetic field at $P (a, 0, a)$ $\vec{B} = \frac{1}{\sqrt{2}} (\hat{i} + \hat{k})$ .

Q 10 :

An infinitely long conductor $P Q R$ is bent to form a right angle as shown in Figure. A current $I$ flows through $P Q R$ . The magnetic field due to this current at the point $M$ is $H_{1}$ . Now, another infinitely long straight conductor $Q S$ is connected at $Q$ so that current is $I / 2$ in $Q R$ as well as in $Q S$ , the current in $P Q$ remaining unchanged. The magnetic field at $M$ is now $H_{2}$ . The ratio $\frac{H_{1}}{H_{2}}$ is given by [2000]

$\frac{1}{2}$
$1$
$\frac{2}{3}$
$2$

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(3)

Magnetic field at $M$ due to $P Q$ and $Q R$

$H_{1} = \frac{1}{2} [\frac{μ_{0} I}{2 π R}] + 0 = \frac{μ_{0} I}{4 π R}$

[ $∵$ Magnetic field B = 0 at any point on the current-carrying straight conductor]

Now when wire $Q S$ is joined,

$H_{2}$ = (Magnetic field at M due to PQ) + (Magnetic field at M due to QR) + (Magnetic field at M due to QS)

$= \frac{1}{2} [\frac{μ_{0} I}{2 π R}] + 0 + \frac{1}{2} [\frac{μ_{0} I / 2}{2 π R}]$ $= \frac{3 μ_{0} I}{8 π R}$

$∴$ $\frac{H_{1}}{H_{2}} = \frac{\frac{μ_{0} I}{4 π R}}{\frac{3 μ_{0} I}{8 π R}} = \frac{2}{3}$

Q 11 :

Two long parallel wires are at a distance $2 d$ apart. They carry steady equal currents flowing out of the plane of the paper, as shown. The variation of the magnetic field $B$ along the line $X X'$ is given by [2000]

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(2)

Here current is flowing out of the plane of the paper, so the magnetic field at points to the right of the wire will be upwards and to the left will be downwards. Let us consider certain points.

Point C (midpoint between A and B): The magnetic field at $C$ due to $A$ $({\vec{B}}_{C A})$ is in the upward direction, but the magnetic field at $C$ due to $B$ is in the downward direction. Hence the net field is zero.

Point E: Magnetic field due to $A$ is upward and magnetic field due to $B$ is downward, but $| {\vec{B}}_{E A} |$ $<$ $| {\vec{B}}_{E B} |$ .

$∴$ Net magnetic field is in the downward direction.

Point D: $| {\vec{B}}_{D A} |$ $>$ $| {\vec{B}}_{D B} |$ . Hence the net magnetic field is upwards. Similarly, other points can be considered.

Hence graph (2) satisfies these conditions.

Q 12 :

A cylindrical cavity of diameter $a$ exists inside a cylinder of diameter $2 a$ as shown in the figure. Both the cylinder and the cavity are infinitely long. A uniform current density $J$ flows along the length. If the magnitude of the magnetic field at the point $P$ is given by $\frac{N}{12} μ_{0} a J,$ then the value of $N$ is [2012]

(5)

$Current density J = \frac{current}{area} = \frac{I}{A} \Rightarrow I = J A$

$Magnetic field B_{R} after removing cavity (C)$

$B_{R} = B_{total} - B_{cavity}$

$B_{R} = \frac{μ_{0} I_{t}}{2 π a} - \frac{μ_{0} I_{c}}{2 π (\frac{3}{2} a)}$

$= \frac{μ_{0}}{π a} [\frac{I_{t}}{2} - \frac{I_{c}}{3}]$ $(here I_{t} = J (π a^{2}), I_{c} = J (\frac{π a^{2}}{4}))$

$= \frac{μ_{0}}{π a} [\frac{π a^{2} J}{2} - \frac{π a^{2} J}{12}]$ or, $B_{R} = \frac{5 μ_{0} a J}{12}$

$Comparing it with \frac{N}{12} μ_{0} a J, we get N = 5$

Q 13 :

A steady current $I$ goes through a wire loop $P Q R$ having the shape of a right angle triangle with $P Q = 3 x, P R = 4 x$ and $Q R = 5 x .$ 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 $k$ . [2009]

(7)

The magnetic field B due to wires PR and PQ = 0. Only wire QR will produce magnetic field at P. From point P, $P M ⊥ Q R .$

$Magnetic field at P due to wire R Q$

$B = \frac{μ_{0}}{4 π} \frac{I}{P M} (\cos θ_{1} + \cos θ_{2})$ ...(i)

$In ∆ P Q M,$ $9 x^{2} = P M^{2} + a^{2}$ ...(ii)

$In ∆ P R M,$ $16 x^{2} = P M^{2} + {(5 x - a)}^{2}$ ...(iii)

$\Rightarrow 16 x^{2} - 9 x^{2} = {(5 x - a)}^{2} - a^{2}$

$\Rightarrow 7 x^{2} = 25 x^{2} - 10 a x$ $\Rightarrow 10 a x = 18 x^{2}$

$\Rightarrow a = 1.8 x$ ...(iv)

From eq. (ii) and (iv),

$9 x^{2} = P M^{2} + {(1.8 x)}^{2}$

$∴$ $P M = \sqrt{9 x^{2} - 3.24 x^{2}} = \sqrt{5.76 x^{2}} = 2.4 x$ ...(v)

Also $\cos θ_{1} = \frac{a}{3 x} = \frac{1.8 x}{3 x} = 0.6$ ...(vi)

$\cos θ_{2} = \frac{5 x - a}{4 x} = \frac{5 x - 1.8 x}{4 x} = \frac{3.2}{4} = 0.8$ ...(vii)

$Therefore, from eq. (i), (v), (vi) and (vii),$

$B = \frac{μ_{0}}{4 π} \times \frac{I}{2.4 x} (0.6 + 0.8)$ $= \frac{μ_{0}}{4 π} \times \frac{I}{2.4 x} \times 1.4$ $= \frac{7 μ_{0} I}{48 π x}$

$Comparing it with B = k (\frac{μ_{0} I}{48 π x}), we get k = 7$

Q 14 :

Two concentric circular loops, one of radius $R$ and the other of radius $2 R$ , lie in the $x y$ -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 $I_{2} > 2 I_{1}$ . $\vec{B} (x, y)$ denotes the magnetic field at a point $(x, y)$ in the $x y$ -plane. Which of the following statement(s) is(are) correct? [2021]

$\vec{B} (x, y)$ is perpendicular to the $x y$ -plane at any point in the plane.
$| \vec{B} (x, y) |$ depends on $x$ and $y$ only through the radial distance $r = \sqrt{x^{2} + y^{2}}$
$| \vec{B} (x, y) |$ is non-zero at all points for $r < R$
$\vec{B} (x, y)$ points normally outward from the $x y$ -plane for all the points between the two loops.

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Select one or more options

(1, 2)

As per Biot--Savart's law,

$d \vec{B} = \frac{μ_{0} i d \vec{ℓ} \times \vec{r}}{4 π r^{3}}$

i.e., $\vec{B}$ is perpendicular to both $i d \vec{ℓ}$ and $\vec{r}$

$d \vec{ℓ}$ is in the $x y$ -plane and $\vec{r}$ is also in the $x y$ -plane.

$∴$ $d \vec{B}$ is perpendicular to the $x y$ -plane.

Due to symmetry, it depends only on the distance from the centre.

Radial distance $r = \sqrt{x^{2} + y^{2}}$

At the centre, $B_{1} = \frac{μ_{0} I_{1}}{2 R}$ and $B_{2} = \frac{μ_{0} I_{2}}{4 R}$

Clearly, $B_{2} > B_{1}$

As we approach towards the first loop, $B_{1}$ increases to infinity; hence $B_{1}$ dominates.

So, it would be zero at some point between the inner loops and the centre.

Field will be in opposite directions inside and outside the loop.

Q 15 :

A steady current $I$ flows along an infinitely long hollow cylindrical conductor of radius R. This cylinder is placed coaxially inside an infinite solenoid of radius 2R. The solenoid has n turns per unit length and carries a steady current $I$ . Consider a point P at a distance $r$ from the common axis. The correct statement(s) is(are): [2013]

In the region $0 < r < R$ , the magnetic field is non-zero.
In the region $R < r < 2 R$ , the magnetic field is along the common axis.
In the region $R < r < 2 R$ , the magnetic field is tangential to the circle of radius $r$ , centered on the axis.
In the region $r > 2 R$ , the magnetic field is non-zero.

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Select one or more options

(1, 4)

In the region $0 < r < R$ , the net magnetic field is due to the current in the solenoid.

In the region $r > 2 R$ , the magnetic field is present due to the current in the cylinder.

For the region $R < r < 2 R$ , the magnetic field is neither along the common axis, nor tangential to the circle of radius $r$ .

Topic Question Set

Q 2 :

Which one of the following options represents the magnetic field $\vec{B}$ at O due to the current flowing in the given wire segments lying on the $x y$ plane? [2022]

Q 3 :

A symmetric star shaped conducting wire loop is carrying a steady state current $I$ as shown in the figure. The distance between the diametrically opposite vertices of the star is $4 a$ . The magnitude of the magnetic field at the center of the loop is [2017]

Q 4 :

An infinitely long hollow conducting cylinder with inner radius $\frac{R}{2}$ and outer radius $R$ carries a uniform current density along its length. The magnitude of the magnetic field, $| \vec{B} |$ as a function of the radial distance $r$ from the axis is best represented by [2012]

Q 5 :

A loop carrying current $I$ lies in the $x - y$ plane as shown in the figure. The unit vector $\hat{k}$ is coming out of the plane of the paper. The magnetic moment of the current loop is [2012]

Q 7 :

A long straight wire along the Z-axis carries a current $I$ in the negative Z-direction. The magnetic vector field $\vec{B}$ at a point having coordinates $(x, y)$ in the $Z = 0$ plane is [2002]

Q 8 :

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 center is [2001]

Q 9 :

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 [2001]

Q 11 :

Two long parallel wires are at a distance $2 d$ apart. They carry steady equal currents flowing out of the plane of the paper, as shown. The variation of the magnetic field $B$ along the line $X X'$ is given by [2000]

Q 13 :

A steady current $I$ goes through a wire loop $P Q R$ having the shape of a right angle triangle with $P Q = 3 x, P R = 4 x$ and $Q R = 5 x .$ 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 $k$ . [2009]

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Topic Question Set

Q 2 : 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? [2022]

Q 3 : A symmetric star shaped conducting wire loop is carrying a steady state current I as shown in the figure. The distance between the diametrically opposite vertices of the star is 4a. The magnitude of the magnetic field at the center of the loop is [2017]

Q 4 : An infinitely long hollow conducting cylinder with inner radius R2 and outer radius R carries a uniform current density along its length. The magnitude of the magnetic field, |B→| as a function of the radial distance r from the axis is best represented by [2012]

Q 5 : A loop carrying current I lies in the x-y plane as shown in the figure. The unit vector k^ is coming out of the plane of the paper. The magnetic moment of the current loop is [2012]

Q 6 : 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 XY plane and a steady current 'I' flows through the wire. The Z-component of the magnetic field at the centre of the spiral is [2011]

Q 7 : 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 coordinates (x,y) in the Z=0 plane is [2002]

Q 8 : 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 center is [2001]

Q 9 : 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 2a. The magnetic field due to this loop at the point P(a,0,a) points in the direction [2001]

Q 11 : Two long parallel wires are at a distance 2d apart. They carry steady equal currents flowing out of the plane of the paper, as shown. The variation of the magnetic field B along the line XX' is given by [2000]

Q 13 : A steady current I goes through a wire loop PQR having the shape of a right angle triangle with PQ=3x, PR=4x and QR=5x. If the magnitude of the magnetic field at P due to this loop is k(μ0I48πx), find the value of k. [2009]

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Q 2 :

Which one of the following options represents the magnetic field $\vec{B}$ at O due to the current flowing in the given wire segments lying on the $x y$ plane? [2022]

Q 3 :

A symmetric star shaped conducting wire loop is carrying a steady state current $I$ as shown in the figure. The distance between the diametrically opposite vertices of the star is $4 a$ . The magnitude of the magnetic field at the center of the loop is [2017]

Q 4 :

An infinitely long hollow conducting cylinder with inner radius $\frac{R}{2}$ and outer radius $R$ carries a uniform current density along its length. The magnitude of the magnetic field, $| \vec{B} |$ as a function of the radial distance $r$ from the axis is best represented by [2012]

Q 5 :

A loop carrying current $I$ lies in the $x - y$ plane as shown in the figure. The unit vector $\hat{k}$ is coming out of the plane of the paper. The magnetic moment of the current loop is [2012]

Q 7 :

A long straight wire along the Z-axis carries a current $I$ in the negative Z-direction. The magnetic vector field $\vec{B}$ at a point having coordinates $(x, y)$ in the $Z = 0$ plane is [2002]

Q 8 :

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 center is [2001]

Q 9 :

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 [2001]

Q 11 :

Two long parallel wires are at a distance $2 d$ apart. They carry steady equal currents flowing out of the plane of the paper, as shown. The variation of the magnetic field $B$ along the line $X X'$ is given by [2000]

Q 13 :

A steady current $I$ goes through a wire loop $P Q R$ having the shape of a right angle triangle with $P Q = 3 x, P R = 4 x$ and $Q R = 5 x .$ 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 $k$ . [2009]