Understanding the Relationship Between Electrical Power, Resistance, Charge, Current, and Time

The question asks for the correct relation connecting electrical power (P), resistance (R), the total charge (Q) flowing through a wire, and the time (t) taken for this charge flow. The options provided also include electric current (I).

To find the correct relationship, we need to use fundamental electrical formulas that relate these quantities. The key concepts are power dissipated in a resistor, Ohm's Law, and the definition of electric current.

Fundamental Electrical Formulas

• Electric current (I) is defined as the rate of flow of electric charge. If charge Q flows through a cross-section of a wire in time t, the current I is given by:

\[I = \frac{Q}{t}\]

• Ohm's Law relates voltage (V), current (I), and resistance (R) in a circuit:

\[V = IR\]

• Electrical power (P) dissipated in a resistor can be expressed in several ways. One common formula related to current and resistance is:

\[P = I^2 R\]

We need to find a relation that involves P, R, Q, and t, potentially also including I as it appears in the options.

Deriving the Correct Relation

Let's start with the power formula that relates power, current, and resistance:

\[P = I^2 R\]

We need to introduce charge (Q) and time (t) into this equation. We know the relationship between current, charge, and time is \(I = \frac{Q}{t}\). While we could substitute I directly, let's try multiplying the power equation by time 't':

\[P \times t = I^2 R \times t\]

\[Pt = I^2 Rt\]

Now, let's rearrange the terms on the right side: \(I^2 Rt = I \times (I \times t) \times R\).

From the definition of current, we know that \(I = \frac{Q}{t}\), which implies that \(I \times t = Q\).

Substitute \(I \times t = Q\) into the equation \(Pt = I \times (I \times t) \times R\):

\[Pt = I \times Q \times R\]

Rearranging the terms on the right side, we get:

\[Pt = IRQ\]

This derived relation connects power (P), time (t), current (I), resistance (R), and charge (Q).

Checking the Options

Let's compare our derived relation \(Pt = IRQ\) with the given options:

• Option 1: \(Pt = IRQ\) - This matches our derived relation.
• Option 2: \(PI = QRt\) - This does not match \(Pt = IRQ\).
• Option 3: \(PQ = IRt\) - This does not match \(Pt = IRQ\).
• Option 4: \(PR = QIt\) - This is equivalent to \(PR = IQt\), which does not match \(Pt = IRQ\).

Therefore, the correct relation is \(Pt = IRQ\).

Let's quickly verify the units. Power (P) is in Watts (W), time (t) is in seconds (s). So Pt has units of Joule (J), which is energy. Current (I) is in Amperes (A), Resistance (R) is in Ohms (\(\Omega\)), Charge (Q) is in Coulombs (C). The unit of \(IRQ\) would be \(A \times \Omega \times C\). From V=IR, \(A \times \Omega\) is Volts (V). So \(IRQ\) has units \(V \times C\). Since Energy = Charge \(\times\) Voltage (\(E=QV\)), the units \(V \times C\) are Joules (J). The units match, adding confidence to the relation.

The derivation shows how the relation \(Pt = IRQ\) arises directly from fundamental principles of electricity relating power, current, resistance, charge, and time.

Revision Table: Key Electrical Formulas

Additional Information: Concepts of Power, Charge, and Resistance

Electric Power (P): Power is the rate at which electrical energy is transferred or dissipated in a circuit. It is measured in Watts (W). In a resistor, electrical energy is converted into heat.

Electric Charge (Q): Charge is a fundamental property of matter. It is carried by particles like electrons (negative charge) and protons (positive charge). The unit of charge is the Coulomb (C).

Electric Current (I): Current is the flow of electric charge. It is typically the flow of electrons in metals. It is measured in Amperes (A). One Ampere is equal to one Coulomb of charge flowing per second (1 A = 1 C/s).

Resistance (R): Resistance is a measure of how much a material opposes the flow of electric current. It is measured in Ohms (\(\Omega\)). Materials with low resistance conduct current easily, while materials with high resistance oppose current flow.

The relation \(Pt = IRQ\) connects the energy dissipated (\(Pt\), since Energy = Power \(\times\) Time) with the circuit parameters (I, R) and the total charge flown (Q). While less common than \(E = I^2 Rt\), it is a valid relation derived from fundamental principles.