## A metallic rod of mass per unit length 0.5 kg m-1 is lying horizontally on a smooth inclined plane which makes an angle of…

Q: A metallic rod of mass per unit length 0.5 kg m-1 is lying horizontally on a smooth inclined plane which makes an angle of 30° with the horizontal. The rod is not allowed to slide down by flowing a current through it when magnetic field of induction 0.25 T is acting on it in the vertical direction .The current flowing in the rod to keep it stationary is

(a) 14.76 A

(b) 11.32 A

(c) 5.98 A

(d) 7.14 A

Ans: (b)

Sol: For stationary rod ,

$\displaystyle i l B cos\theta = m g sin\theta$

$\displaystyle i = \frac{m g sin\theta}{l B cos\theta}$

$\displaystyle i = \frac{m g tan\theta}{l B }$

$\displaystyle i = \frac{0.5 \times 10 \times tan30}{1 \times 0.25 }$

i = 11.32 A

## A proton, a deutron and an α – particle with the same K.E. enter a region of uniform magnetic field, moving at right angle to B…

Q: A proton, a deutron and an α – particle with the same K.E. enter a region of uniform magnetic field, moving at right angle to B. What is the radio of the radius of their circular paths?

(a) 1 : √2: 1

(b) 1 : √2 : √2

(c) √2 : 1 : 1

(d) √2 : √2 : 1

Ans: (a)

Sol: In a magnetic field, perpendicular to velocity of particle

$\displaystyle \frac{q v B}{} = \frac{m v^2}{r}$

$\displaystyle r = \frac{m v}{q B}$

Kinetic Energy $\displaystyle E = \frac{1}{2} m v^2 = \frac{(m v)^2}{2 m}$

$\displaystyle r = \frac{\sqrt{2 m E}}{q B}$

$\displaystyle r \propto \frac{\sqrt{m}}{q}$ (For constant value of E & B)

$\displaystyle r_p : r_d : r_{\alpha} = \frac{\sqrt{m_p}}{q_p } : \frac{\sqrt{m_d}}{q_d } : \frac{\sqrt{m_{\alpha}}}{q_{\alpha} }$

$\displaystyle r_p : r_d : r_{\alpha} = \frac{\sqrt{m}}{e} : \frac{\sqrt{2m}}{e} : \frac{\sqrt{4 m}}{2e }$

= 1 : √2: 1

## The magnetic field normal to the plane of a wire of n turns and radius r which carries a current I is measured on the axis…

Q: The magnetic field normal to the plane of a wire of n turns and radius r which carries a current I is measured on the axis of the coil at a small distance h from the center of the coil. This is smaller than the magnetic field at the center by the fraction

(a) $\displaystyle \frac{2}{3}\frac{r^2}{h^3}$

(b) $\displaystyle \frac{3}{2}\frac{r^2}{h^3}$

(c) $\displaystyle \frac{2}{3}\frac{r^2}{h^2}$

(d) $\displaystyle \frac{3}{2}\frac{h^2}{r^2}$

Ans: (d)

Sol: Magnetic field at the centre is

$\displaystyle B_1 = \frac{\mu_0}{4\pi} \frac{2\pi n I}{r}$

Magnetic field on the axis is

$\displaystyle B_2 = \frac{\mu_0}{4\pi} \frac{2\pi n I r^2}{(r^2 + h^2)^{3/2}}$

On dividing

$\displaystyle \frac{B_2}{B_1} = (1+\frac{h^2}{r^2})^{-3/2}$

Fractional decrease in the magnetic field will be

$\displaystyle \frac{B_1 – B_2}{B_1} = (1 – \frac{B_2}{B_1})$

$\displaystyle = 1- (1+\frac{h^2}{r^2})^{-3/2}$

$\displaystyle = 1- (1- \frac{3}{2}\frac{h^2}{r^2})$

$\displaystyle = \frac{3}{2}\frac{h^2}{r^2}$

## Equal currents are flowing in three infinitely long wires along positive x, y and z directions. The magnetic field at …..

Q. Equal currents are flowing in three infinitely long wires along positive x, y and z directions. The magnetic field at a point (0, 0, -a) would be (i = current in each wire )

(a) $\displaystyle \frac{\mu_0 i}{2\pi a} (\hat{j}-\hat{i})$

(b) $\displaystyle \frac{\mu_0 i}{2\pi a} (\hat{i}-\hat{j})$

(c) $\displaystyle \frac{\mu_0 i}{2\pi a} (\hat{j}+\hat{i})$

(d) $\displaystyle \frac{\mu_0 i}{2\pi a} (-\hat{i}-\hat{j})$

Ans: (a)

## A tightly-wound long solenoid has n turns per unit length, radius r and carries a current i. A particle having …..

Q. A tightly-wound long solenoid has n turns per unit length, radius r and carries a current i. A particle having charge q and mass m is projected from a point on the axis in the direction perpendicular to the axis. The maximum speed for which particle does not strike the solenoid will be

(a) $\displaystyle \frac {\mu_0 q r n i}{2m}$

(b) $\displaystyle \frac {\mu_0 q r n i}{m}$

(c) $\displaystyle \frac {2\mu_0 q r n i}{3m}$

(d) None of these

Ans: (a)

Sol: Magnetic field inside the solenoid is B = μ0 n i , along the axis of solenoid

Since velocity is Perpendicular to B , hence particle moves in a circle of radius

$\displaystyle R = \frac{m v}{q B} = \frac{m v}{q \mu_0 n i}$

If particle does not strike solenoid

R ≤ r/2

$\displaystyle \frac{m v}{q \mu_0 n i} \le \frac{r}{2}$

$\displaystyle v \le \frac{\mu_0 n i q r}{2 m}$

$\displaystyle v_{max} = \frac{\mu_0 n i q r}{2 m}$