anteena gain in db? link budeget equation for tx system?received power on earth?

A geosynchronous satellite is located above the earth in the magnetosphere at a distance of 40, 000km transmits a TV signal of 6Watt power to an earth-based receiver using a dish antenna of diameter 0.5m. The diameter of the antenna of the satellite earth receiver is 5m. If the downlink frequency is 4GHz and the antenna aperture is 60%
a) what is the antenna gains in dB
b) What is the link budget equation for this transmission system?
c) what is the received power on earth?


please give me the formulas and help me in the problem

You have mentioned that the antenna aperture is 60%, I would assume that you meant the antenna aperture efficiency \[\eta _e \] to be 60%.

The effective aperture \[A_e\] of the antenna is linked to the antenna gain \[G\] and its physical aperture \[A_p\] via:
\[A_e = \frac{\lambda^2 G}{4\pi} = \eta _e A_p\]

From the above equation, you can calculate the gain

After this, you can come up with the link budget equation as:
\[ P_r = \frac{G_t G_r \lambda^2}{(4 {\pi} R)^2} \frac{1}{L_s L_f}P_t \]

In your case, since they are not mentioned, the losses in the middle term could be ignored.

Gain of transmitting antenna \[G_t\]
Gain of receiving antenna \[G_r\]
Distance of separation \[R\]
Transmit power \[P_t\]
Received power \[P_r\]
Pointing error, atmospheric attenuation, polarization loss \[L_s\]
Margin loss due to fading and other effects \[L_f\]