You are in the lab with a 10 V battery. You have a 10 W light bulb and you want to put 0.75 A of current through the light bulb and 7.5 V across the light bulb. This will allow the light bulb to have a brightness proportional to 5.625 W. Assuming any resistors that you need, design and build a circuit that delivers the required current and voltage to the bulb

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To design a circuit that delivers the required current (0.75 A) and voltage (7.5 V) to the light bulb, we need to understand Ohm's Law and power calculations.

Ohm's Law states that the current passing through a conductor between two points is directly proportional to the voltage across the two points, and inversely proportional to the resistance between them. Mathematically, it can be represented as:

V = I * R

Where:
V is the voltage across the conductor (in volts)
I is the current passing through the conductor (in amperes)
R is the resistance of the conductor (in ohms)

Power can be calculated using the formula:

P = V * I

Where:
P is the power (in watts)
V is the voltage (in volts)
I is the current (in amperes)

Based on the given requirements, we know the current I (0.75 A) and voltage V (7.5 V) we want to achieve. We can rearrange the Ohm's Law equation to solve for the resistance R required:

R = V / I

R = 7.5 V / 0.75 A
R = 10 ohms

Therefore, to design the circuit, we need a resistor with a resistance of 10 ohms in series with the light bulb.

Here's a circuit diagram of the setup:

```
+-----------------------------+
10 V Battery ---->| |
| |
| 10 ohm Resistor |
| |
+----+------------------------+
|
---
|
|
+----+-------------------------+
Light Bulb ------>| |
| |
+-----------------------------+
```

Connect the positive terminal of the battery to one end of the resistor, and the other end of the resistor to one side of the light bulb. Finally, connect the other side of the light bulb back to the negative terminal of the battery.

This circuit configuration will provide the required current of 0.75 A and the voltage of 7.5 V across the light bulb, allowing it to have a brightness proportional to 5.625 W.