Question

The voltage across the $$10\mathrm{\Omega }$$ resistor in the given circuit is x volt. Then find the value of x to the nearest integer.

Answer 1

Hint: In order to proceed with the problem first we need to know about ohm’s law. It states that the voltage across a conductor is directly proportional to the current flowing through it.

Formula Used:
To find the equation for voltage we have,
$$V=IR$$
Where, I is current and R is resistance.

Complete step by step solution:

Image: A circuit consists of three resistors and a battery.

In the above circuit, the voltage across the $$10\mathrm{\Omega }$$ resistor in the given circuit is x volt. We need to find the value of x to the nearest integer. From ohm’s law, we have,
$$I={\displaystyle \frac{V}{R}}$$
$$\Rightarrow V=IR$$…………. (1)

In order to find the current in the circuit, we need to simplify the given circuit by finding the equivalent resistance of the two resistors given in a parallel combination.

Image: An equivalent resistor.

The equivalent resistance is given by,
$$R_{eq}={\displaystyle \frac{R_2{R}_3}{R_2+R_3}}$$
Here, $$R_2=50\mathrm{\Omega }$$ and $$R_3=20\mathrm{\Omega }$$
Substitute the value in the above equation we get,
$$R_{eq}={\displaystyle \frac{50\times 20}{50+20}}$$
$$\Rightarrow R_{eq}={\displaystyle \frac{1000}{70}}\mathrm{\Omega }$$
$$\Rightarrow R_{eq}={\displaystyle \frac{100}{7}}\mathrm{\Omega }$$

Then,
$$R=R_{eq}+R_1$$
Here, $$R_1=10\mathrm{\Omega }$$
Then the above equation will become,
$$R={\displaystyle \frac{100}{7}}+10$$
$$\Rightarrow R={\displaystyle \frac{100+70}{7}}$$
$$\Rightarrow R={\displaystyle \frac{170}{7}}$$
$$\Rightarrow R=7\mathrm{\Omega }$$

Using Ohm’s law,
$$I={\displaystyle \frac{V}{R}}$$
Now substitute the value of V and R in the above equation we get,
$$I={\displaystyle \frac{170}{7}}$$
$$\Rightarrow I=7A$$
Now, in order to find the voltage across $$10\mathrm{\Omega }$$ resistor, consider equation (1)
$$V=IR$$
$$\Rightarrow V=7\times 10$$
$$\therefore V=70V$$

Therefore, the value of x is 70.

Note:Here it is important to remember that, in a circuit when two are more resistors are connected in parallel, we need to apply equivalent resistance formula as $$R_{eq}={\displaystyle \frac{R_2{R}_3}{R_2+R_3}}$$ and if the resistors are connected in series, then we need to add all the resistors to get equivalent value of resistors.