Question

hos 5. A long solid right circular cylinder of radius R carries a current I, which is uniformly distributed. Find the magnetic field everywhere, both inside and outside the cylinder. 6. A long solenoid of area A = TtR,? and turns per unit length n carries a current i = lycoswt. Find the electric field at a distance r from the axis of the solenoid. Distinguish between the cases r > R: and r < Rs. 7. A lens of focal length 10cm is used to produce an image with magnification +3. What must the object distance be? 8. Slits of width d are carved into a screen. For what range of slit separations will there be 11 interference fringes within the central diffraction maximum?

only 5

Answer 1

5. Inside the cyllinder, Ampere's Law can be written as

$\oint \bar{B}\cdot \bar{dl}=\mu _{0}I_{enc}$

$B2\pi r=\mu _{0}I\frac{\pi r^{2}}{\pi R^{2}}$

$B_{in} =\frac{\mu _{0}Ir}{2\pi R^{2}}$

And for the outer region

$B2\pi r=\mu _{0}I$

$B_{out}=\frac{\mu _{0}I}{2\pi r}$

6. For points inside the solenoid we can write Faraday's Law as

$\oint \bar{E}\cdot \bar{dl}=-\frac{d\Phi _{B}}{dt}$

$E2\pi r=-\pi r^{2}\frac{dB}{dt}$

Where the B field inside the solenoid is

$B=\mu _{0}nI$

Therefore

$\frac{dB}{dt}=\mu _{0}n\frac{dI}{dt}=-\mu _{0}nI_{0}\omega \sin \omega t$

Replacing back we get

$E_{in}= \frac{r\mu _{0}nI_{0}\omega \sin \omega t}{2}$

For points outside

$E2\pi r=-\pi R^{2}_{1}\frac{dB}{dt}$

$E_{out}=\frac{R^{2}_{1}\mu _{0}nI_{0}\omega \sin \omega t}{2 r}$

7. We use the lens equation and the magnification formula:

$d_{o}=\frac{fd_{i}}{d_{i}-f}$

$-\frac{d_{i}}{d_{o}}=3$

Combining this two

$d_{o}=\frac{f(-3d_{o})}{(-3d_{o})-f}$

Solving for d_o we get

$d_{o}=\frac{2}{3}f=\frac{20}{3}cm$

8. The formula for diffraction grating is

$d\sin \theta =n\lambda$

Where d is the slit spacing. The condition for 11 interference fringes within the central maximum is

$d\sin 90^{\circ} =11\lambda$

$d =11\lambda$