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The gap between the electrodes in an automobile engine's spark plug is \(0.060 \mathrm{~cm}\). To produce an electric spark in a gasolineair mixture, an electric field of \(3.0 \times 10^{6} \mathrm{~V} / \mathrm{m}\) must be achieved. On starting a car, what minimum voltage must be supplied by the ignition circuit? Assume the field is uniform.(answer check available at lightandmatter.com)

Short Answer

Expert verified
The minimum voltage required is 1800 V.

Step by step solution

01

Understand the Relationship between Electric Field and Voltage

The electric field between two points can be defined as the voltage difference divided by the distance between the points. Mathematically, this is expressed as \( E = \frac{V}{d} \), where \( E \) is the electric field, \( V \) is the voltage, and \( d \) is the distance between the electrodes.
02

Identify Known Values

In this problem, we know that the electric field \( E \) is \( 3.0 \times 10^6 \) V/m. The gap distance \( d \) is given as \( 0.060 \) cm. First, convert this distance to meters for consistent units, so \( d = 0.060 \) cm = \( 0.00060 \) m.
03

Rearrange the Formula to Solve for Voltage

To find the voltage \( V \), rearrange the formula: \( V = E \times d \). This will allow us to calculate the required minimum voltage.
04

Calculate the Minimum Voltage Required

Substitute the known values into the rearranged formula: \( V = (3.0 \times 10^6 \text{ V/m}) \times 0.00060 \text{ m} = 1800 \text{ V}.\)
05

Interpret the Result

The calculation shows that the minimum voltage required to produce the necessary electric field and create a spark across the gap is \( 1800 \) volts.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Voltage Calculation
Voltage is crucial in numerous electric applications, especially when aiming to generate a spark in a spark plug. Essentially, voltage refers to the electric potential difference between two points. When calculating the voltage necessary to create an electric spark, we use the relationship between the electric field, voltage, and distance. This is represented by the formula: \[ V = E \times d \]Here, \( V \) stands for voltage, \( E \) represents the electric field, and \( d \) is the distance, often called the spark plug gap in this context. By understanding this relationship, one can calculate the minimum voltage needed to generate a spark by knowing the electric field strength and the gap distance.
Spark Plug Gap
The spark plug gap is a critical feature in an engine's ignition system. This gap is the distance that the electric spark needs to jump to ignite the air-fuel mixture within an engine cylinder. In our example, the gap is \( 0.060 \) cm, which is the equivalent of \( 0.00060 \) meters when converted to the typical unit used for such calculations.When the ignition circuit supplies adequate voltage, an electric spark is generated across this gap. It's vital for the spark plug gap to be set correctly鈥攖oo narrow a gap can lead to weak sparks, while too wide a gap may prevent sparking altogether. Thus, maintaining the right gap is essential for vehicle performance and efficiency.
Uniform Electric Field
A uniform electric field means that the electric field strength is consistent at every point between two electrodes. This uniformity is essential in the functioning of a spark plug, where it ensures that the electric field is strong enough to produce a spark that can jump across the spark plug gap.In our specific case, a uniform electric field of \( 3.0 \times 10^{6} \) V/m is needed to produce the spark. This uniformity aids in achieving reliable ignition by providing a steady energy front that can ionize the air in the gap efficiently, making the process predictable and repeatable.
Electric Field Formula
The electric field formula is integral for many calculations related to electricity and magnetism. This formula is given as:\[ E = \frac{V}{d} \]Where \( E \) is the electric field in volts per meter (V/m), \( V \) is the voltage in volts (V), and \( d \) is the distance between the points in meters (m).By rearranging this formula, we can solve for any one of these variables if the other two are known. In our case, rearranging it to find \( V \) gives us the vital equation used for determining the voltage required for sparking across the gap:\[ V = E \times d \]Understanding this relationship is key to solving problems involving electric fields and voltage differences, particularly in applications like spark plugs where precise calculations ensure proper engine performance.

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Most popular questions from this chapter

Calculate the quantity \(i^{i}\) (i.e., find its real and imaginary parts).(answer check available at lightandmatter.com)

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