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**2. the intensity of light bulb A increases. **

The light bulbs in the circuit are identical. When the switch is closed, 1. both go out. 2. the intensity of light bulb A increases. 3. the intensity of light bulb A decreases. 4. the intensity of light bulb B increases. 5. the intensity of light bulb B decreases. 6. some combination of 1–5 occurs. 7. nothing changes. Answer: 7. Because the light bulbs are identical, the potential difference across each is 12 V, and so nothing happens with the switch is closed.

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The circuit photographed below includes three identical light bulbs and two identical batteries, wired as seen in the photograph. In this photograph the switch is OFF, that is, the circuit is open at that point. The intensity level of the two light bulbs is identical (well, almost in the photograph). What will happen when the switch is turned ON, closing the circuit. The upper light bulb will: (1) become brighter. (2) become dimmer. (3) stay the same. The lower light bulb will: (1) become brighter. (2) become dimmer. (3) stay the same. The center light bulb will: (1) go ON. (2) stay OFF. ∆𝑉=0

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**Example: Determine the current through each resistor.**

Junction: I1 I2 I Loops: Loop 1 Loop 1 Loop 2 Loop 2 Solve for I2 using loop 2: Solve for I1 using loop 1: Solve for I using Junction rule: The negative signs tell us we chose the wrong direction. Current flows in the opposite direction.

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Now that we have discussed resistors and capacitors separately, it is time to put them together in the same circuit. Circuits that contain resistors and capacitors in the same circuit are more commonly called RC circuits. What happens when the switch is open? No current can move through the circuit and therefore nothing happens in the circuit. What happens once the switch is closed? s Current leaves the battery, passes through the resistor and charges up the capacitor. What would happen if the battery was replaced with a wire? Charge would flow out of the capacitor and cause a current in the circuit, which would flow in the opposite direction. The current in an RC circuit will vary with time while the capacitor is charging or discharging. When charging the current will start at a maximum value and decrease over time until it reaches zero. When discharging the current will start at a maximum value and decrease over time until it reaches zero.

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**Let us find a mathematical model to describe this situation.**

Using Kirchhoff’s Laws: Voltage across capacitor Capacitor stores energy, but circuit loses energy. Voltage across resistor Resistor uses energy What is the magnitude of the current the instant the switch is closed? There is no charge initially on the capacitor, therefore there is zero potential difference across the capacitor and the current will be defined by the resistor. Maximum current that will be supplied by the battery. How much charge is stored on the capacitor after a long time? After a long time the capacitor is full and no more current can flow into the capacitor. Maximum amount of charge that can be stored on the capacitor.

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**How then does the current change between the initial time and a long time later?**

First order separable differential equation q = 0 when t = 0 and at a time t the charge is q. This tells us how the charge changes with time while we are Charging a Capacitor.

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