Consider the three circuits shown below. All the resistors and all the batteries are identical. Which of the statements are true and which ones are false?

1. The power dissipated in circuit A is twice the power dissipated in circuit B.
2. The current through the resistor is the same for circuits A and C.
3. The current through the resistor is the same for circuits A and B.
4. The voltage at one resistor in circuit C is twice the voltage at one resistor in circuit B.
5. The total energy dissipated in circuit C is twice the total energy dissipated in circuit B.

The question aims to answer statements about the three circuits given above. Difference between potential in two points in an electric field, which helps current to flow in the circuits, is called voltage (V). The term current is defined as the rate of flow of electrons in the circuit.

Expert Answer

Part(a)

Yes, the statement $(a)$ is true. The energy dissipated in circuit $A$ is twice as large as in-circuit $B$. The current though $A$ is twice as much as the current to $B$, so it provides a dispersed power twice the size, assuming that both circuits have the same power source.

Part(b)

Yes, statement $(b)$ is correct. Circuit $C$ is a different type of circuit compared to $A$. The current through resistors is the same; however, for each circuit, the current need for the source in each circuit is different. Circuit $A$ requires $\dfrac{1}{2}$ of the current in the source compared to its $C$ counterpart.

For circuit $A$, current is calculated by using following procedure.

\[I=\dfrac{V}{R}\]

Let’s suppose $V=10v$ and $R=1\Omega$

\[I=\dfrac{10}{1}=10 A\]

For circuit $C$, the current is calculated by using the following procedure. There are two branches, so there are two values of current.

\[I_{1}=\dfrac{V}{R_{1}}\]

\[I_{2}=\dfrac{V}{R_{2}}\]

Let’s suppose $V=10v$, $R_{1}=1\Omega$ and $R_{2}=1\Omega$

\[I_{1}=\dfrac{10}{1}=10 A\]

\[I_{2}=\dfrac{10}{1}=10 A\]

\[I=I_{1}+I_{2}\]

\[I=20A\]

The current in the resistor is the same in both circuits, but the overall current is different.

Part(c)

Yes, the statement is correct. In circuit $B$, the current flow is the same in each resistor in the circuit, and in this case, since they are the same resistance, the voltage through each resistor is $\dfrac{1}{2}V$.

Part(d)

Yes, the statement is correct. Voltage along a single resistor in circuit $C$ is twice as compared to circuit $B$. Circuit $B$ is a series circuit, so voltage divides across two resistors.

Part(e)

Yes, the $IV$ current strength in $C$ is twice as high as the current in $B$. So, the statement is correct.

Numerical Result

(a) The statement is correct.

(b) The statement is correct.

(c) The statement is correct.

(d) The statement is correct.

(e) The statement is correct.

Example

Consider the two circuits shown below. All resistors and all batteries are the same. Which statements are true and which are false?

– The energy dissipated in circuit $B$ is twice the force dissipated in circuit $A$.

Solution

No, statement $(a)$ is not true. The energy dissipated in circuit $A$ is twice as large as in-circuit $B$. The current through $A$ is twice as much as the current to $B$, so it provides a dispersed power twice the size, assuming that both circuits have the same power source. Hence, the statement is not true.

Images/Mathematical drawings are created with Geogebra.

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