Unit 2: Divider Quest - Subjective Questions

Voltage divider law states that the voltage across any resistor in a series circuit is proportional to its resistance. For a series circuit containing a supply voltage and resistors , the voltage across is given by The law can be applied when: - The circuit elements are connected in series. - The same current flows through every resistor. - The resistors have finite and known values. - The output load is either absent or has been included in the equivalent resistance calculation. The law is useful for obtaining a required fraction of the input voltage without using a separate voltage source.

For two resistors connected in series, the total resistance is Since the same current flows through both resistors, Ohm's law gives The voltage across is Therefore, Similarly, the voltage across is Hence, Adding both voltages gives , which verifies Kirchhoff's Voltage Law.

The voltage divider law can be validated using the following procedure: - Connect resistors and in series across a DC supply voltage . - Measure the supply voltage using a voltmeter. - Measure the voltage across and across . - Calculate the theoretical values using - Compare the measured and calculated voltages. - Verify that the measured values approximately satisfy . Small differences may occur because of resistor tolerance, supply variation, voltmeter loading, and measurement errors. Close agreement validates the voltage divider law.

Let , , and . The total resistance is Voltage across is Voltage across is Therefore, the voltages are across the resistor and across the resistor. The sum is , confirming the result.

When a load resistor is connected across the output resistor , it appears in parallel with . The effective output resistance becomes The loaded output voltage is therefore Since is smaller than , the loaded output voltage is lower than the no-load value. This effect is called loading. Loading can be reduced by using a load resistance much larger than the divider resistance or by using a buffer amplifier with high input impedance.

Unloaded voltage divider: - No external load is connected across the output. - The output voltage is calculated directly from the divider resistors. - For two resistors, . - The output voltage is independent of an external load. Loaded voltage divider: - A load resistance is connected across the output resistor. - The output resistor and load form a parallel combination. - The output voltage is calculated using the equivalent resistance. - The output voltage is generally lower because the load draws current. The loaded case represents practical circuits more accurately.

For a two-resistor divider, the output across is Therefore: - Increasing increases the fraction of the supply voltage appearing across . - Increasing decreases the output voltage across . - If , the output voltage is half the supply voltage. - If , the output approaches the supply voltage. - If , the output approaches zero. The output depends on the ratio of the resistors, not only on their absolute values, provided loading is negligible.

Consider resistors connected in series across a voltage source . The total resistance is The current through the series circuit is The voltage across resistor is Substituting the current gives Hence, the general voltage divider equation is This shows that each resistor receives a voltage proportional to its resistance.

Current divider law states that the current entering a parallel network divides among the branches in inverse proportion to their resistances. For two parallel resistors, the current through is and the current through is The law applies when the branches are connected in parallel, so that the same voltage appears across each branch. The branch currents must add to the total current according to Kirchhoff's Current Law:

Since and are in parallel, they have the same voltage . By Ohm's law, The total current is The equivalent resistance is Therefore, the common voltage is . The current through is Similarly, Thus, current divides inversely with resistance.

The current divider law can be validated as follows: - Connect resistors and in parallel. - Connect the parallel combination to a DC source through a circuit that supplies total current . - Measure the total current entering the parallel network. - Measure the branch currents and using ammeters. - Calculate the theoretical values using - Compare the measured and calculated values. - Check that . Agreement within experimental tolerance validates the current divider law.

Let , , and . Current through is Current through is The lower resistance carries the larger current. Verification gives

| Feature | Voltage Divider Law | Current Divider Law |\n|---|---|---|\n| Circuit arrangement | Series resistors | Parallel resistors |\n| Common quantity | Current is common to all elements | Voltage is common to all branches |\n| Divided quantity | Supply voltage | Total current |\n| Division relationship | Voltage divides directly in proportion to resistance | Current divides inversely in proportion to resistance |\n| Basic two-element expression | | |\n| Verification law | | | Both laws are derived from Ohm's law and Kirchhoff's laws.

In a parallel circuit, every branch has the same voltage across it. According to Ohm's law, branch current is Since the voltage is fixed, current is inversely proportional to resistance: Consequently, a branch with smaller resistance offers less opposition to charge flow and carries more current. For example, if two branches have resistances and , their currents are Thus, the branch with resistance carries twice the current of the branch with resistance .

Let resistors be connected in parallel across a common voltage . The current through resistor is The total current is the sum of all branch currents: Therefore, Substituting this into the branch-current equation gives Thus, the current in a branch is proportional to its conductance and inversely related to its resistance.

For equal resistors, the division is uniform. In a series circuit containing equal resistors, each resistor has resistance , so the voltage across each resistor is In a parallel circuit containing equal resistors, the total current divides equally, so the current in each branch is Therefore: - Equal series resistors divide voltage equally. - Equal parallel resistors divide current equally. - These results assume ideal connections and identical resistor values.

Practical measurements may differ from theoretical values because of several factors: - Resistor tolerance: A resistor marked as may have an actual value within its specified tolerance range. - Voltmeter loading: A voltmeter has finite input resistance and may draw current from a voltage divider, changing the output voltage. - Ammeter resistance: An ammeter has a small internal resistance that can alter branch resistance and current. - Supply variation: The source voltage or current may fluctuate during measurement. - Lead and contact resistance: Wires and connections introduce small additional resistances. - Instrument accuracy: Resolution and calibration limits create measurement uncertainty. Validation should therefore compare measured and theoretical values using percentage error:

The theoretical output voltage is The measured output is . The percentage error is Therefore, the theoretical output is and the measurement has a error. This small difference may be caused by resistor tolerance, supply variation, or voltmeter loading.

Let , , and . Current through is Current through is Therefore, the branch carries and the branch carries . The lower resistance carries the greater current because both branches have the same voltage and current is inversely proportional to resistance. The result satisfies .

Divider laws are direct applications of fundamental circuit laws: - Ohm's law: Relates voltage, current, and resistance using . - Kirchhoff's Voltage Law: In a series circuit, the sum of voltage drops equals the source voltage: Combining this law with Ohm's law gives the voltage divider law. - Kirchhoff's Current Law: At a parallel junction, the incoming current equals the sum of outgoing branch currents: Combining this law with Ohm's law and the common branch voltage gives the current divider law. Thus, divider laws are convenient forms of Kirchhoff's laws for resistor networks.