You have a special light bulb with a very delicate wire filament. The wire will break if the current in it ever exceeds 1.50 A, even for an instant. What is the largest root-mean-square current you can run through this bulb?

Verified step by step guidance

1. Understand that the root-mean-square (RMS) current is a measure of the effective current value for alternating current (AC) circuits. It is related to the peak current by the formula: <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>I</mi><mi>rms</mi></msub><mo>=</mo><mfrac><msub><mi>I</mi><mi>peak</mi></msub><mi>&#x221A;2</mi></mfrac></math>.
2. Identify that the peak current <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>I</mi><mi>peak</mi></msub></math> is the maximum current that can flow through the filament without breaking it. In this case, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>I</mi><mi>peak</mi></msub></math> is 1.50 A.
3. Use the formula for RMS current to find the largest RMS current that can safely flow through the bulb: <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>I</mi><mi>rms</mi></msub><mo>=</mo><mfrac><mn>1.50</mn><mi>&#x221A;2</mi></mfrac></math>.
4. Calculate the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x221A;2</mi></math>, which is approximately 1.414, to use in the formula.
5. Divide the peak current by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x221A;2</mi></math> to find the RMS current: <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>I</mi><mi>rms</mi></msub><mo>=</mo><mfrac><mn>1.50</mn><mn>1.414</mn></mfrac></math>.

Key Concepts

Root-Mean-Square (RMS) Current

The root-mean-square (RMS) current is a measure of the effective value of an alternating current (AC), which is equivalent to the direct current (DC) that would deliver the same power to a resistor. It is calculated as the square root of the average of the squares of the instantaneous current values over one cycle. RMS current is crucial for determining safe operating levels in AC circuits.

Alternating Current (AC)

Alternating current (AC) is a type of electrical current in which the flow of electric charge periodically reverses direction. Unlike direct current (DC), which flows in one direction, AC is used in most household and industrial applications due to its ability to be easily transformed to different voltages. Understanding AC is essential for calculating RMS values and ensuring safe operation of electrical devices.

Current Limitation in Electrical Components

Electrical components, such as light bulb filaments, have a maximum current rating beyond which they can be damaged or destroyed. This limitation is due to the heat generated by the current, which can exceed the material's tolerance. Knowing the maximum allowable current is vital for preventing damage and ensuring the longevity of electrical devices, especially delicate ones like the filament in the question.