Problem 62 - Physics for Scientists and Engineers with Modern Physics

Question

The primary windings of a transformer which has an \(85 \%\) efficiency are connected to 110-V ac. The secondary windings are connected across a \(2.4-\Omega, 75-\mathrm{W}\) lightbulb. (a) Calculate the current through the primary windings of the transformer. (b) Calculate the ratio of the number of primary windings of the transformer to the number of secondary windings of the transformer.

Step-by-Step Solution

Answer

(a) The current through the primary windings is calculated using the primary power and voltage. (b) The primary to secondary turns ratio is the primary voltage divided by the secondary voltage.

Step 1: Calculate the Secondary Current

The secondary windings connect to a 75-W lightbulb with a resistance of 2.4 Ω. Use Ohm's Law \( I = \frac{P}{V} \) to find the secondary current \( I_s \). The power \( P = 75 \) W and the resistance \( R = 2.4 \) Ω: \[ I_s = \frac{75}{\sqrt{75 \times 2.4}} \] Substitute the values to calculate \( I_s \).

Step 2: Secondary Voltage Calculation

Use Ohm’s Law \( V = IR \) to find the secondary voltage \( V_s \). With \( I_s \) from Step 1 and \( R = 2.4 \) Ω, \[ V_s = I_s \times R \].

Step 3: Calculate the Primary Power

Since the efficiency of the transformer is 85%, the primary power \( P_p \) is given by \( P_p = \frac{P_s}{0.85} \) where \( P_s = 75 \) W. Substitute to find \( P_p \).

Step 4: Calculate the Primary Current

Use the formula \( I_p = \frac{P_p}{V_p} \), where \( V_p = 110 \) V. With \( P_p \) from Step 3, calculate \( I_p \).

Step 5: Calculate the Turns Ratio

The turns ratio \( n \) is given by \( n = \frac{V_p}{V_s} \). Use \( V_s \) from Step 2 and \( V_p = 110 \) V to calculate \( n \).

Key Concepts

Ohm's Law

Ohm's Law is a fundamental concept in electrical engineering that relates voltage, current, and resistance through the formula: \[ V = IR \]where \( V \) is the voltage across an element in volts, \( I \) is the current through it in amperes, and \( R \) is the resistance in ohms. Understanding this law is pivotal for analyzing circuits and devices like transformers.In simple terms, Ohm's Law states that the current flowing through a conductor between two points is directly proportional to the voltage across the two points. This principle helps us calculatethe current across various components of an electrical circuit safe and efficient design and usage.- **Power and Ohm's Law:** Power \( P \) in electrical terms is the rate at which energy is consumed or generated. It can be related to the other components as: \[ P = VI \] - Using substitution in Ohm's Law, another variation is used: \[ P = I^2R \] or \[ P = \frac{V^2}{R} \]This allows us to solve for any variable given the other two, making Ohm's Law incredibly versatile in electrical calculations, including finding the secondary current in our circuit.

Primary and Secondary Current

In transformers, the current through the primary and secondary windings plays a crucial role in determining the transformer’s performance and efficiency. The primary current (\( I_p \)) and the secondary current (\( I_s \)) are linked by the power relationship, assuming ideal conditions, such that the input power equals the output power, modified slightly by transformer losses.- **Primary Current (\( I_p \))**: It is the current entering the primary side of the transformer. It can be calculated using transformer efficiency and power equations. - Given the primary voltage \( V_p \) and the transformed power \( P_p \), the primary current is calculated as: \[ I_p = \frac{P_p}{V_p} \]- **Secondary Current (\( I_s \))**: This is the current delivered by the secondary side, it ties directly into the resistive load connected, in this case, a lightbulb. Using Ohm's Law when power \( P \) and resistance \( R \) are known: \[ I_s = \frac{P}{V_s} \]Understanding these components allows us to manage transformer outputs safely and determine exact conditions for various applications.

Turns Ratio

The turns ratio of a transformer is a critical concept that defines how the voltages and currents are transformed between the primary and secondary windings. This ratio directly determines the voltage transformation from one circuit to another.- **Definition**: The turns ratio \( n \) in a transformer is the ratio of the number of turns in the primary coil \( N_p \) to the number of turns in the secondary coil \( N_s \), \[ n = \frac{N_p}{N_s} = \frac{V_p}{V_s} \]Here, the voltage ratio \( \frac{V_p}{V_s} \) is used assuming the transformer is ideal, meaning no losses.- **Utility of Turns Ratio**: - **Voltage Transformation**: It dictates how the input voltage is modified into output voltage, crucial for matching loads with supplies. - **Current Adjustment**: While voltage is transformed proportionally to the turns ratio, current is inversely proportional. As voltage steps up, current steps down proportionately, and vice versa. Understanding the turns ratio helps in designing transformers for the desired voltage and current outcomes for various technological applications.