Using current technology (and by that I mean experiments and telescopes that are available now) we would probably be unable to detect radio signals from Earth even if observed from a distance of a few light years. Therefore there is currently no prospect of detecting such signals from (around) another star.
If we are talking about detecting "Earth", and assuming that we are not talking about deliberate beamed attempts at communication, then we must rely on detecting random radio "chatter" and accidental signals generated by our civilisation.
The SETI Phoenix project was the most advanced search for radio signals from other intelligent life. Quoting from Cullers et al. (2000): "Typical signals, as opposed to our strongest signals fall below the detection threshold of most surveys, even if the signal were to originate from the nearest star". Quoting from Tarter (2001): "At current levels of sensitivity, targeted microwave searches could detect the equivalent power of strong TV transmitters at a distance of 1 light year (within which there are no other stars)...". The equivocation in these statements is due to the fact that we do emit stronger beamed signals in certain well-defined directions, for example to conduct metrology in the solar system using radar. Such signals have been calculated to be observable over a thousand light years or more. But these signals are brief, beamed into an extremely narrow angle and unlikely to be repeated. You would have to be very lucky to be observing in the right direction at the right time if you were performing targeted searches.
Hence my assertion that with current methods and telescopes there is not much chance of success. But of course technology advances and in the next 10-20 years there may be better opportunities.
I did read (struggling to locate it now) that SETI were targeting multiple transiting exoplanets. The idea here is that you can wait for the two planets in the transiting planets to come into line and then hope that some beamed signal from the inner planet to the outer planet "spills over" and heads towards the Earth - ingenious.
It has been suggested that new radio telescope projects and technology like the Square Kilometre Array may be capable of serendipitously detecting radio "chatter" out to distances of 50 pc ($\sim 150$ light years) - see Loeb & Zaldarriaga (2007). This array, due to begin full operation some time after 2025 could also monitor a multitude of directions at once for beamed signals. A good overview of what might be possible in the near future is given by Tarter et al. (2009).
Loeb & Zaldarriga give a table (Table 1) in their paper that lists some sources of transmission from the Earth (which are the basis for their claim). The most powerful pseudo-isotropic sigmals appear to come in the 40-850 MHz range from TV transmitters, with a summed power of $10^{9}$ W or $10^{10} $W/Hz.
Disclaimer, I'm no expert on the details, but I know the general idea.
Large ground telescopes are great, and often superior to orbital telescopes. The reason is as you said - it's cheaper to build ground telescopes, which means for the same budget you can build a bigger, more powerful ground telescope. It's true that space-based telescopes don't have to deal with the Earth's atmosphere obscuring things, but then there are also so-called adaptive optics that mitigate this advantage.
Three of the biggest advantages of ground telescopes are:
- You can make them very large. Because of the way optical resolution works ($\theta = 1.22 \lambda/D$), big telescopes have a fundamental advantage over small telescopes, and you can make bigger ground telescopes for the same price. You can actually see this in your numbers. The James Webb telescope has a 7m mirror, while the Thirty Meter Telescope is four times as big.
- In the same way, because they are larger, they can collect more light in a given amount of time. To get the same amount of light with a smaller telescope, you need to observe for longer, which is bad (telescope time is at a premium in astronomy; most of the time astronomers need to apply for time).
- They are easy to repair. If something in the Hubble Space Telescope breaks, you might need to send up astronauts and conduct a space walk, which is obviously very expensive. Comparatively even the most inaccessible ground locations (like the South Pole) can be reached for a fraction of the price of going to space.
The discrepancy is actually such that many space probes aren't worth funding because one should just build a bigger, more powerful ground telescope.
Given the above one could flip the question around and ask, why bother with space-based telescopes then? There are reasons, some of the most important being:
- The atmosphere obscures some wavelengths of light. If you want to observe in those wavelengths you must go to space.
- Ground telescopes are susceptible to local weather conditions. If it's raining or cloudy, you can't observe.
- Space telescopes can observe in all directions all the time. On the ground, you can only observe at night, and even then you can only observe half the celestial sphere at best (because the Earth is in the way of the other half).
See e.g. this source for more details.
Best Answer
It's spherical because the main dish cannot be steered; steering is done by moving the receiver (the big thing hanging over the center of the reflector). A parabolic reflector would produce varying errors when aimed in different directions; a spherical reflector has the same error for all directions. Presumably the receiver is designed to compensate for this.
Source: Wikipedia.