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Magazine Chapter 33: Problem 14
Maximum Charge In an oscillating \(L C\) circuit, \(L=1.10 \mathrm{mH}\) and \(C=4.00 \mu \mathrm{F}\). The maximum charge on the capacitor is \(3.00 \mu \mathrm{C}\). Find the maximum current.
Short Answer
Expert verified
The maximum current is approximately 0.143 A.
Step by step solution
01
- Understand the Problem
The problem involves finding the maximum current in an LC circuit given the values of the inductance (L), the capacitance (C), and the maximum charge (Q) on the capacitor.
02
- Write Down Given Values
Given values are: Inductance, \(L = 1.10 \text{ mH} = 1.10 \times 10^{-3} \text{ H}\) Capacitance, \(C = 4.00 \text{ 渭F} = 4.00 \times 10^{-6} \text{ F}\) Maximum charge, \(Q_{\text{max}} = 3.00 \text{ 渭C} = 3.00 \times 10^{-6} \text{ C}\)
03
- Use the Energy Conservation Principle
In an LC circuit, the total energy is conserved and it can be expressed either as the energy stored in the capacitor or the energy stored in the inductor. Therefore, \[ \frac{1}{2} L I_{\text{max}}^2 = \frac{1}{2} \frac{Q_{\text{max}}^2}{C} \]
04
- Isolate the Maximum Current
Solve for the maximum current, \(I_{\text{max}}\), by rearranging the equation: \[ L I_{\text{max}}^2 = \frac{Q_{\text{max}}^2}{C} \]\[ I_{\text{max}}^2 = \frac{Q_{\text{max}}^2}{L C} \]\[ I_{\text{max}} = \frac{Q_{\text{max}}}{\text{sqrt}(LC)} \]
05
- Substitute the Values
Substitute the given values into the equation:\[ I_{\text{max}} = \frac{3.00 \times 10^{-6}}{\text{sqrt}((1.10 \times 10^{-3})(4.00 \times 10^{-6}))} \] Calculate the denominator first: \[ (1.10 \times 10^{-3}) \times (4.00 \times 10^{-6}) = 4.40 \times 10^{-9} \] Then take the square root: \[ \text{sqrt}(4.40 \times 10^{-9}) = 2.10 \times 10^{-5} \]
06
- Compute the Final Value
Now divide the maximum charge by the square root value: \[ I_{\text{max}} = \frac{3.00 \times 10^{-6}}{2.10 \times 10^{-5}} \] Calculate the final value: \[ I_{\text{max}} \thickapprox 0.143 \text{ A} \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Inductance
Inductance, often denoted as L, is a property of an electrical circuit that resists changes in the current flowing through it. Think of it as inertia for electric current. In an LC circuit, which consists of an inductor (L) and a capacitor (C), the inductance plays a critical role in determining the oscillation period of the circuit. Measured in henries (H), inductance depends on factors like the number of coils in the inductor, the core material, and the coil's geometry. For example, in our exercise, we have an inductor with an inductance of 1.10 mH (millihenries).
Capacitance
Capacitance, symbolized as C, is the ability of a system to store an electric charge. Capacitors, which are components that display capacitance, store energy in the electric field created between two conductive plates separated by an insulating material. The unit of capacitance is the farad (F). In our LC circuit, the given capacitance is 4.00 渭F (microfarads). Capacitance affects how much charge can be stored and how the circuit oscillates. It's important to note that in any LC circuit, the capacitance and inductance together dictate the natural frequency of oscillation.
Maximum Charge
The maximum charge (Q_max) on a capacitor is the highest amount of electric charge it can hold at any given point in time. In the exercise, the maximum charge on the capacitor is given as 3.00 渭C (microcoulombs). This value is crucial because it will determine the energy stored in the capacitor when it is fully charged. Using this charge, we can find other parameters like the maximum current in the circuit through the principles of energy conservation.
Maximum Current
The maximum current (I_max) in an LC circuit is the highest amount of current that can flow through the inductor when the energy stored in the capacitor is fully converted to magnetic energy in the inductor. Once you know the maximum charge and the values for inductance and capacitance, you can use the formula derived from energy conservation: \[ I_{\text{max}} = \frac{Q_{\text{max}}}{\sqrt{LC}} \]In our example, the given values would be substituted to find the maximum current, which approximately equals 0.143 A (amperes). This shows the peak current that the oscillating circuit will reach during its cycle.
Energy Conservation
Energy conservation in an LC circuit states that the total energy remains constant. This energy oscillates between the capacitor and the inductor. When the capacitor is fully charged, all the energy is stored as electric energy: \[ \frac{1}{2} \frac{Q_{\text{max}}^2}{C} \]As the capacitor discharges, this energy converts into magnetic energy stored in the inductor: \[ \frac{1}{2} L I_{\text{max}}^2 \]These two expressions help us understand that at any point in the oscillation cycle, the sum of the electric energy in the capacitor and the magnetic energy in the inductor is constant. This principle is the basis for solving problems related to LC circuits.
