Answer :
To solve this problem, we need to find the total resistance of three resistors connected in parallel and then determine the current when a 12 V battery is connected.
### Step 1: Calculate Total Resistance
When resistors are connected in parallel, the formula for total resistance ([tex]\(R_{\text{total}}\)[/tex]) is given by:
[tex]\[
\frac{1}{R_{\text{total}}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}
\][/tex]
Here, [tex]\(R_1\)[/tex] is 2 ohms, [tex]\(R_2\)[/tex] is 3 ohms, and [tex]\(R_3\)[/tex] is 6 ohms. We can substitute these values into the formula:
[tex]\[
\frac{1}{R_{\text{total}}} = \frac{1}{2} + \frac{1}{3} + \frac{1}{6}
\][/tex]
Calculating each term:
- [tex]\( \frac{1}{2} = 0.5 \)[/tex]
- [tex]\( \frac{1}{3} \approx 0.333 \)[/tex]
- [tex]\( \frac{1}{6} \approx 0.167 \)[/tex]
Add them up:
[tex]\[
\frac{1}{R_{\text{total}}} = 0.5 + 0.333 + 0.167 = 1.0
\][/tex]
Thus,
[tex]\[
R_{\text{total}} = \frac{1}{1.0} = 1.0 \, \text{ohm}
\][/tex]
### Step 2: Calculate Total Current
To find the total current ([tex]\(I_{\text{total}}\)[/tex]) flowing through the circuit, we use Ohm's Law which states:
[tex]\[
I = \frac{V}{R}
\][/tex]
Where [tex]\(V\)[/tex] is the voltage (12 V in this case) and [tex]\(R\)[/tex] is the total resistance we calculated:
[tex]\[
I_{\text{total}} = \frac{12}{1.0} = 12 \, \text{amperes}
\][/tex]
Therefore, the total resistance of the resistors connected in parallel is 1.0 ohms, and the current flowing through the circuit with a 12 V battery is 12 amperes.
### Step 1: Calculate Total Resistance
When resistors are connected in parallel, the formula for total resistance ([tex]\(R_{\text{total}}\)[/tex]) is given by:
[tex]\[
\frac{1}{R_{\text{total}}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}
\][/tex]
Here, [tex]\(R_1\)[/tex] is 2 ohms, [tex]\(R_2\)[/tex] is 3 ohms, and [tex]\(R_3\)[/tex] is 6 ohms. We can substitute these values into the formula:
[tex]\[
\frac{1}{R_{\text{total}}} = \frac{1}{2} + \frac{1}{3} + \frac{1}{6}
\][/tex]
Calculating each term:
- [tex]\( \frac{1}{2} = 0.5 \)[/tex]
- [tex]\( \frac{1}{3} \approx 0.333 \)[/tex]
- [tex]\( \frac{1}{6} \approx 0.167 \)[/tex]
Add them up:
[tex]\[
\frac{1}{R_{\text{total}}} = 0.5 + 0.333 + 0.167 = 1.0
\][/tex]
Thus,
[tex]\[
R_{\text{total}} = \frac{1}{1.0} = 1.0 \, \text{ohm}
\][/tex]
### Step 2: Calculate Total Current
To find the total current ([tex]\(I_{\text{total}}\)[/tex]) flowing through the circuit, we use Ohm's Law which states:
[tex]\[
I = \frac{V}{R}
\][/tex]
Where [tex]\(V\)[/tex] is the voltage (12 V in this case) and [tex]\(R\)[/tex] is the total resistance we calculated:
[tex]\[
I_{\text{total}} = \frac{12}{1.0} = 12 \, \text{amperes}
\][/tex]
Therefore, the total resistance of the resistors connected in parallel is 1.0 ohms, and the current flowing through the circuit with a 12 V battery is 12 amperes.
