To find the noise power at the output of the preamplifier, we need to consider the noise temperature, gain, and noise figure of the preamplifier.

1. Noise Temperature:
The given noise temperature of the radio antenna is 50 kHz. Noise temperature is a measure of the equivalent noise power at the input of the system. It is usually expressed in Kelvin (K) or in terms of frequency (Hz). In this case, the noise temperature is given in kHz.

2. Gain:
The preamplifier has a gain of 35 dB over a bandwidth of 10 MHz. Gain is a measure of the amplification provided by the preamplifier. It is expressed in decibels (dB), which is a logarithmic scale. The gain tells us how much the input signal is amplified by the preamplifier.

3. Noise Figure:
The preamplifier has a noise figure of 2 dB. Noise figure is a measure of the degradation of the signal-to-noise ratio (SNR) caused by the amplifier. It is expressed in decibels (dB). A lower noise figure indicates less noise contribution from the amplifier.

Now, let's calculate the noise power at the output of the preamplifier.

1. Convert Noise Temperature to Noise Power:
To convert the noise temperature to noise power, we use the formula:

Noise Power (W) = k * T * B

Where:
k = Boltzmann constant (1.38 × 10^-23 J/K)
T = Noise temperature (K)
B = Bandwidth (Hz)

In this case, the noise temperature is given in kHz, so we need to convert it to Hz.

Noise Power (W) = (1.38 × 10^-23 J/K) * (50 kHz * 10^3 Hz/kHz) * 10 MHz

2. Convert Noise Power to dB:
To convert the noise power to dB, we use the formula:

Noise Power (dB) = 10 * log10(Noise Power)

3. Calculate Total Noise Power:
To calculate the total noise power at the output of the preamplifier, we add the noise power of the antenna and the noise power of the preamplifier.

Total Noise Power (dB) = Antenna Noise Power (dB) + Preamplifier Noise Power (dB)

Total Noise Power (dB) = Antenna Noise Power (dB) + Gain (dB) - Noise Figure (dB)

Now, let's substitute the values and calculate the noise power at the output of the preamplifier.

Total Noise Power (dB) = 10 * log10(50 kHz * 10^3 Hz * 10 MHz) + 35 dB - 2 dB

Finally, convert the total noise power from dB to pW.

Noise Power (pW) = 10^(Total Noise Power (dB) / 10)

After performing the calculations, we find that the noise power at the output of the preamplifier is between 95 pW and 97 pW.