Question

LC-circuit of the radio receiver consists of variable capacitor (Cmin= 1 pF, Cmax=10 pF) and inductor

Answer 1

Answer:

the radio can tune wavelengths between 1.88 and 5.97 m

Explanation:

The signal that can be received is the one that is in resonance as the impedance of the LC circuit.

         X = X_c - X_L

         X = 1 / wC - w L

at the point of resonance the two impedance are equal so their sum is zero

         X_c = X_L

         1 / wC = w L

         w² = 1 / CL

         w = $\sqrt{\frac{1}{CL} }$

let's look for the extreme values

C = 1  10⁻¹² F

         w = $\sqrt{\frac{1}{ 1 \ 10^{-12} \ 1 \ 10^{-6}} }$

         w = $\sqrt{1 \ 10^{18}}$

         w = 10⁹ rad / s

C = 10 10⁻¹² F

         w = $\sqrt{\frac{1}{10 \ 10^{-12} \ 1 \ 10^{-6}} }$Ra 1/10 10-12 1 10-6

         w = $\sqrt{0.1 \ 10^{18}}$Ra 0.1 1018

         w = 0.316 10⁹ rad / s

Now the angular velocity and the frequency are related

           w = 2π f

           f = w / 2π

the light velocity  is

           c = λ f

           λ = c / f

we substitute

          λ = c 2π/w

               

we calculate the two values

 C = 1 pF

          λ₁ = 3 10⁸ 2π / 10⁹

          λ₁= 18.849 10⁻¹ m

          λ₁ = 1.88 m

C = 10 pF

           λ₂ = 3 10⁸ 2π / 0.316 10⁹

           λ₂ = 59.65 10⁻¹ m

           λ₂ = 5.97 m

so the radio can tune wavelengths between 1.88 and 5.97 m