Bob of a simple pendulum of length $$l$$ is made of iron. The pendulum is oscillating over a horizontal coil carrying direct current. If the time period of the pendulum is T then :

The bob of the pendulum is made of iron, and it is oscillating over a horizontal coil carrying direct current. The direct current in the coil produces a constant magnetic field. Since the coil is horizontal, the magnetic field along its axis is vertical. Iron is a ferromagnetic material, so it is attracted to the magnetic field. The coil is below the pendulum, so the magnetic force on the iron bob acts downward, toward the coil.

The gravitational force on the bob is $$ mg $$ downward. The additional magnetic force $$ F_m $$ also acts downward. Therefore, the total downward force when the bob is at rest is $$ mg + F_m $$. The effective acceleration due to gravity, $$ g_{\text{eff}} $$, is given by:

$$ g_{\text{eff}} = g + \frac{F_m}{m} $$

where $$ m $$ is the mass of the bob. Since $$ F_m $$ is positive and downward, $$ g_{\text{eff}} > g $$.

The time period $$ T $$ of a simple pendulum depends on the effective acceleration due to gravity:

$$ T = 2\pi\sqrt{\frac{l}{g_{\text{eff}}}} $$

Because $$ g_{\text{eff}} > g $$, we have:

$$ \frac{l}{g_{\text{eff}}} < \frac{l}{g} $$

Taking square roots (all terms are positive):

$$ \sqrt{\frac{l}{g_{\text{eff}}}} < \sqrt{\frac{l}{g}} $$

Multiplying both sides by $$ 2\pi $$:

$$ 2\pi\sqrt{\frac{l}{g_{\text{eff}}}} < 2\pi\sqrt{\frac{l}{g}} $$

So:

$$ T < 2\pi\sqrt{\frac{l}{g}} $$

Now, consider damping. In air alone, damping occurs due to air resistance. With the coil present, the constant magnetic field and the motion of the iron bob induce eddy currents in the bob. By Lenz's law, these eddy currents oppose the motion of the bob, causing additional electromagnetic damping. Therefore, the total damping (air resistance plus electromagnetic damping) is larger than in air alone.

Hence, the time period is less than $$ 2\pi\sqrt{\frac{l}{g}} $$ and damping is larger than in air alone.

Comparing with the options:

Option A states $$ T < 2\pi\sqrt{\frac{l}{g}} $$ but damping is smaller, which is incorrect.

Option B states $$ T = 2\pi\sqrt{\frac{l}{g}} $$ and damping is larger, which is incorrect.

Option C states $$ T > 2\pi\sqrt{\frac{l}{g}} $$ and damping is smaller, which is incorrect.

Option D states $$ T < 2\pi\sqrt{\frac{l}{g}} $$ and damping is larger, which matches our conclusion.

Hence, the correct answer is Option D.