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Radar systems often use an important component called an LC circuit, which helps to oscillate signals at very high frequencies. Understanding the frequency of a radar transmitter is crucial. The frequency of oscillation is represented by the letter \( f \), and in the context of this exercise, we are given a frequency of \( 9.00 \text{ GHz} \). To provide a context, one GHz, or gigahertz, equals one billion cycles per second. This means that the radar transmitter's LC circuit completes nine billion cycles each second.

• The frequency is key to determining other properties of the circuit.
• Frequency is inversely proportional to the capacitance and inductance values which are core components of an LC circuit.
• High frequencies, like those in radar applications, help in achieving precise and rapid signal detection.

Understanding this frequency allows us to calculate how the other elements in the circuit will behave, particularly how the LC circuit itself will realize resonance.

Inductance is a property of the coil within an LC circuit, vital for achieving resonance at a specified frequency. To find the required inductance \( L \) for resonance, we utilize the resonance formula, \[ f = \frac{1}{2\pi \sqrt{LC}}\] In simpler terms, this equation shows that the frequency \( f \) is dependent on both the inductance \( L \) and the capacitance \( C \). Solving for \( L \) gives us: \[ L = \frac{1}{(2\pi f)^2 C} \] By plugging in the specific values – a frequency of \( 9.00 \text{ GHz} \) and capacitance of \( 2.00 \text{ pF} \) – we can determine the correct inductance needed. Remember, 1 GHz equals \( 10^9 \text{ Hz} \) and 1 pF equals \( 10^{-12} \text{ F} \).

• First, convert all units into standard SI units.
• Next, substitute these values into our formula to get the inductance.
• The results will be in henries, the unit of inductance.

This calculation ensures that our circuit oscillates exactly at the desired frequency.

Inductive reactance determines how much a coil will oppose the change in current flowing through it at a specific frequency. To compute inductive reactance \( X_L \) in the LC circuit, we employ the formula: \[ X_L = 2 \pi f L \] Here, \( f \) is again the frequency in hertz, and \( L \) is the inductance in henries. Once we have found the ideal inductance for resonance, we can calculate the inductive reactance at the radar frequency of \( 9.00 \text{ GHz} \). The resulting inductive reactance is measured in ohms, providing insight into how much the inductor resists changes at this particular frequency.

• Calculate reactance by first determining \( L \) using the relevant formula for resonance.
• Substitute \( L \) and \( f \) into the reactance equation.
• The outcome, in ohms, represents the circuit's opposition towards alternating current changes.

Understanding inductive reactance is key, especially for optimizing the operation of high-frequency applications like those in radar systems.