Question Detail

Here are the steps to solve this type of problem step-by-step:   Step 1: Recall the Resonant Frequency Formula The resonant frequency $f$ of an $LC$ circuit depends on inductance $L$ and capacitance $C$:   $$f = \frac{1}{2\pi \sqrt{L C}}$$ Step 2: Express the New Values in Terms of the Old Ones

• Both $L$ and $C$ are reduced by a factor of 5:   $$L' = \frac{L}{5}$$ $$C' = \frac{C}{5}$$

Step 3: Set Up the Formula for the New Resonant Frequency ($f'$) Substitute $L'$ and $C'$ into the original formula:   $$f' = \frac{1}{2\pi \sqrt{\left(\frac{L}{5}\right) \left(\frac{C}{5}\right)}}$$ Step 4: Simplify the Denominator

• Multiply inside the square root: $\frac{L}{5} \times \frac{C}{5} = \frac{LC}{25}$
• Take the square root of the denominator: $\sqrt{\frac{LC}{25}} = \frac{\sqrt{LC}}{5}$

Step 5: Relate the New Frequency ($f'$) to the Original Frequency ($f$) Simplify the fraction:   $$f' = 5 \times \left( \frac{1}{2\pi \sqrt{LC}} \right) = 5 \times f$$ Step 6: Calculate the Final Answer

• Multiply your initial resonant frequency ($1000\text{ kHz}$) by $5$ to get the new frequency!