Problem 61

Analysis

This problem tests our understanding of the conditions for complex numbers to be equal and the calculation of the modulus of a complex number. From the given condition, we can find the values of $a$ and $b$, and thus calculate the modulus of the complex number.

Step-by-step Solution

1. Since $a+2i=2-bi$, we can equate the real and imaginary parts on both sides of the equation. This gives us $a=2$ and $b=-2$.

2. Now, we can substitute these values back into the complex number to get $a+bi=2-2i$.

3. The modulus of a complex number $a+bi$ is given by $\sqrt{a^2+b^2}$. So, we have $|a+bi|=|2-2i|=\sqrt{2^2+(-2)^2}=\sqrt{8}=2\sqrt{2}$.

Therefore, the answer is $\boxed{2\sqrt{2}}$.