Magnetic Field Due to Finite Wire

Q. An infinitely long thin non-conducting wire is parallel to the $z$-axis and carries a uniform line charge density $\lambda$. It pierces a thin non-conducting spherical shell of radius $R$ in such a way that the arc $PQ$ subtends an angle ${120}^{\circ }$ at the centre $O$ of the spherical shell, as shown in the figure. The permittivity of free space is ${\in }_0$. Which of the following statements is (are) true?

1. The electric flux through the shell is $\sqrt{3}R\lambda /{\in }_0$
2. The $z$-component of the electric field is zero at all the points on the surface of the shell
3. The electric flux through the shell is $\sqrt{2}R\lambda /{\in }_0$
4. The electric field is normal to the surface of the shell at all points

Q. The magnetic field at the centre of an equilateral triangular loop of side $2L$ and carrying a current $i$ is

1. $\frac{9{\mu }_{0}i}{4\pi L}$
2. $\frac{3\sqrt{3}{\mu }_{0}i}{4\pi L}$
3. $\frac{2\sqrt{3}{\mu }_{0}i}{\pi L}$
4. $\frac{3{\mu }_{0}i}{4\pi L}$

Q.

A short bar magnet of magnetic moment 5.25 × 10⁻² J T⁻¹ is placed with its axis perpendicular to the earth’s field direction. At what distance from the centre of the magnet, the resultant field is inclined at 45º with earth’s field on

(a) its normal bisector and (b) its axis. Magnitude of the earth’s field at the place is given to be 0.42 G. Ignore the length of the magnet in comparison to the distances involved.

Q.

Two identical moving coil galvanometers have $10\Omega$ resistance and full scale deflection at $2\mu A$ current. One of them is converted into a voltmeter of $100$ $mV$ full scale reading and the other into an Ammeter of $1$ $mA$ full scale current using appropriate resistors. These are then used to measure the voltage and current in the Ohm's law experiment with $R=1000\Omega$ resistor by using an ideal cell. Which of the following statement(s) is/are correct?

1. The measured value of $R$ will be $978\Omega <R<982\Omega$
2. The resistance of the ammeter will be $0.02\Omega$ (round off to $2^{nd}$ decimal place)
3. If the ideal cell is replaced by a cell having internal resistance of $5\Omega$ then the measured value of $R$ will be more than $1000\Omega$
4. The resistance of the voltmeter will be $100k\Omega$

Q. A current i is flowing in a straight conductor of length L. The magnetic field at a point distant $\frac{L}{4}$ from its centre will be

1. $\frac{4{\mu }_{0}i}{2\pi L}$
   
2. $\frac{{\mu }_{0}i}{\sqrt{2}L}$
3. $\frac{4{\mu }_{0}i}{\sqrt{5}\pi L}$
   
4. $Zero$

Q.

Find the magnetic field B at the center of a rectangular loop of length l and width b, carrying a current i.

Q. A wire is bent in the form of a circular arc with a straight portion AB. Magnetic field at O when current $I$ is flowing in the wire is

1. $\frac{{\mu }_{0}I}{2r}(\pi -\theta +\tan \theta )$
2. $\frac{{\mu }_{0}I}{2\pi r}(\pi +\theta -\tan \theta )$
3. $\frac{{\mu }_{0}I}{2\pi r}(\pi -\theta +\tan \theta )$
4. $\frac{{\mu }_{0}I}{2\pi r}(-\tan \theta +\pi -\theta )$

Q. The magnetic field at the origin due to the current flowing in the wire is

1. $\frac{{\mu }_{0}I}{8\pi a}(-\hat{i}+\hat{k})$
2. $\frac{{\mu }_{0}I}{8\pi a}(\hat{i}+\hat{k})$
3. $\frac{{\mu }_{0}I}{2\pi a}(\hat{i}+\hat{k})$
4. $\frac{{\mu }_{0}I}{4\pi a\sqrt{2}}(\hat{i}-\hat{k})$

Q. Currents i and 3i are flowing in opposite directions in two co-axial long thin walled circular cylinders of radii a and 2a respectively as shown. Choose correct options. (Length of the cylinder is L)

1. Pressure on inner cylinder wall is zero.
2. Pressure on inner cylinder wall is ${\mu }_0\frac{i^2}{8{\pi }^2{a}^2}$.
3. Pressure on outer cylinder wall is $3{\mu }_0\frac{i^2}{32{\pi }^2{a}^2}$.
4. Force on unit length of outer cylinder is $\frac{3}{4}\frac{{\mu }_0{i}^2}{\pi a}$.

Q. Shown in the figure, is a conductor carrying current $I$. The net magnetic induction at the point $\text{O}$ is -

[Point $\text{O}$ is the common center of all the three arcs.]

1. Zero
2. $\frac{{\mu }_{0}I\theta }{24\pi R}$
3. $\frac{11{\mu }_{0}I\theta }{24\pi R}$
4. $\frac{5{\mu }_{0}I\theta }{24\pi R}$

Q. Figure shows a square loop ABCD with edge length a. The resistance of the wire ABC is r and that of ADC is 2r. The value of magnetic field at the centre of the loop assuming uniform wire is

1. $\frac{\sqrt{2}{\mu }_{0}i}{3\pi a}\otimes$
   
2. $\frac{\sqrt{2}{\mu }_{0}i}{3\pi a}\odot$
   
3. $\frac{\sqrt{2}{\mu }_{0}i}{\pi a}\odot$
   
4. $\frac{\sqrt{2}{\mu }_{0}i}{\pi a}\otimes$

Q. Figure shows an equilateral triangular frame of side L. Frame starts entering into the region of uniform magnetic field B with constant velocity v. If i is instantaneous current through the frame, then the correct graph is. Take anticlockwise current direction to be positive.

1. 
2. 
3. 
4. 

Q. Current i flows through a long conducting wire bent at right angle as shown in figure. The magnetic field at a point P on the right bisector of the angle XOY at a distance r from O is

1. $\frac{2{\mu }_{0}i}{\pi r}$
   
2. $\frac{{\mu }_{0}i}{4\pi r}(\sqrt{2}+1)$
   
3. $\frac{{\mu }_0}{4\pi }.\frac{2i}{r}(\sqrt{2}+1)$
4. $\frac{{\mu }_{0}i}{\pi r}$
   

Q. A closed circuit is in the form of a regular hexagon of side a. If the circuit carries current I, What is magnetic induction at the cetre of the hexagon?

1. $\frac{\sqrt{3}{\mu }_{0}i}{\pi a}$
2. $\frac{\sqrt{3}{\mu }_{0}i}{\pi a}$
3. zero
4. None

Q. $\text{Statement A :}$ A current carrying wire is placed parallel to the external magnetic field. The force on it due to the external magnetic field is zero.

$\text{Statement R :}$ The net charge on current wire is zero.

1. Both $A$ and $R$ are true, and $R$ is the correct explanation of $A$.
2. Both $A$ and $R$ are true but $R$ is not correct explanation of $A$.
3. $A$ is true, but $R$ is false.
4. $A$ is false, but $R$ is true.

Q.

A current I flows in a long thin walled cylinder of radius R. What pressure do the walls of the cylinder experience?

1. $\frac{{\mu }_{0}I}{8\pi R}$
   

2. $\frac{{\mu }_0{I}^2}{8{\pi }^2{R}^2}$
   

3. $\frac{{\mu }_0{I}^2}{16\pi R^2}$
   

4. $\frac{{\mu }_{0}I}{4{\pi }^{2}R}$
   

Q. A point charge of magnitude q = 4.5 nC is moving with speed v = $3.6\times {10}^7$ m/s parallel to the x-axis along the line y = 3 m. Find the magnetic field at the origin produced by this charge when the charge is at the point x = -4 m, y = 3 m, as shown in the figure.

1. $-3.89\times {10}^{-10}\hat{k}T$
2. $3.89\times {10}^{-10}\hat{i}T$
3. $3.89\times {10}^{-10}\hat{j}T$
4. $3.89\times {10}^{-10}\hat{k}T$

Q. Three circular coils each having two turns and radius $R$ are placed mutually perpendicular to each other and each having its centre at the origin of the coordinate system. If current $i$ is flowing through each coil, then the magnitude of the magnetic field at the common centre is:

1. $\frac{\sqrt{3}{\mu }_{0}i}{R}$
2. Zero
3. $\frac{\sqrt{2-1}{\mu }_{0}i}{2R}$
4. $(\sqrt{3}-\sqrt{2})\frac{{\mu }_{0}i}{2R}$

Q. Determine the magnitude of magnetic field at the center $\left(P\right)$ of a regular hexagon shaped current loop of side $L$.

1. $\frac{\sqrt{3}{\mu }_{0}i}{\pi L}$
2. $\frac{3\sqrt{3}{\mu }_{0}i}{\pi L}$
3. $\frac{3{\mu }_{0}i}{\pi L}$
4. $\frac{{\mu }_{0}i}{3\pi L}$

Q. In a certain region, uniform electric field exists as $\overrightarrow{E}=E_0\hat{j}.$ A proton and an electron are projected from origin at time $t=0$ with certain velocities along $+x$-axis direction. Due to the electric field, they experience force and move in the $xy$-plane along different trajectories.

$\left(i\right)$ The path followed by the particles will be a

1. parabola
2. circular
3. hyperbola
4. spiral

Q. A charge of $3.14\times {10}^{-6}\text{C}$ is distributed uniformly over a circular ring of radius $20\text{cm}$. The ring rotates about its axis with angular velocity of $60\text{rad/s}$. Find the ratio of electric field to the magnetic field at a point on the axis located at a distance of $5\text{cm}$ from centre.

1. $1.88\times {10}^{15}\text{m/s}$
2. $2.50\times {10}^{15}\text{m/s}$
3. $3.20\times {10}^{15}\text{m/s}$
4. $4.50\times {10}^{15}\text{m/s}$

Q.

Figure (35-E4) shows a long wire bent at the middle to form a right angle.Show that the magnitudes of the magnetic fields at the points P, Q, R and S are equal and find this magnitude.

Q. A wire carrying current $i$ has the configuration as shown in the figure. Two semi-infinite straight sections, each tangent to the same circle, are connected by a circular arc, of angle $\theta ,$ along the circumference of the circle, with all sections lying in the same plane. What must $\theta (\text{in rad) be in order for}B$ to be zero at the center of circle is

Q. If the magnetic field at ‘P’ can be written as $K$ $\tan \left(\frac{\alpha }{2}\right)$, then $K$ is

1. $\frac{{\mu }_{0}I}{4\pi d}$
2. $\frac{{\mu }_{0}I}{2\pi d}$
3. $\frac{{\mu }_{0}I}{\pi d}$
4. $\frac{2{\mu }_{0}I}{4\pi d}$

Q. A uniform but time varying magnetic field $B=2t^3+24tT$ is present in a cylindrical region of radius $R=2.5cm$ as shown.
The force on an electron at $P$ at $t=2sec$. is

1. $48\times {10}^{-21}N$
2. $24\times {10}^{-21}N$
3. Zero
4. $96\times {10}^{-21}N$

Q.

Figure (35-E6) shows a square loop of edge a made of a uniform wire.A current i enters the loop at the point A and leaves it at the point C.Find the magnetic field at the point P which is on the perpendicular bisector of AB at a distance a/4 from it.

Q.

Consider a straight piece of length x of a wire carrying a current i.Let P be a point on the perpendicular bisector of the piece, situated at a distance d from its middle point.Show that for d>>x, the magnetic field at P varies as 1/$d^2$ whereas d<<x, it varies as 1/d.

Q. Earths magnetic field at a place where angle of dip is 37 degree and vertical component of Earths magnet is 0.5 G is

Q. Figure shows an infinitely long current carrying wire bent at an angle $\alpha$. The magnetic induction at the point $\text{P}$ located on the angle bisector at a distance $x$ from the point $\text{O}$ at the bend is given by $\frac{{\mu }_{0}I}{2\pi x}\cot \frac{\alpha }{n}$. The value of $n$ is

Q. a particle moving towards east enters a magntic field directed towards north and deflected towards vertically downward direction . the charged particled is?