A total charge Q is uniformly distributed on a thread of length ℓ. The thread forms a semicircle. What is the potential at the full-circle’s center? (Assume V = 0 at large distances.)
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Electric Potential
Problem 71b
Textbook Question
An analog voltage signal can vary from 0 V to 5.00 V, and it is to be converted to an 8-bit binary representation. What binary number would best represent 3.47 volts?

1
Step 1: Understand the problem. The analog voltage signal ranges from 0 V to 5.00 V, and it is being converted to an 8-bit binary representation. An 8-bit binary number can represent 2^8 = 256 discrete levels. Each level corresponds to a specific voltage range.
Step 2: Calculate the resolution of the analog-to-digital converter (ADC). The resolution is the smallest voltage increment that can be represented by one binary step. This is given by dividing the voltage range by the number of levels: Resolution = (5.00 V - 0 V) / 256.
Step 3: Determine the binary level corresponding to 3.47 V. To do this, divide the given voltage (3.47 V) by the resolution calculated in Step 2. This gives the level number corresponding to 3.47 V.
Step 4: Round the level number to the nearest integer, as binary representation corresponds to discrete levels. This integer represents the binary level that best approximates 3.47 V.
Step 5: Convert the rounded level number into an 8-bit binary representation. Use the standard method of converting a decimal number to binary, ensuring the result is padded with leading zeros to make it 8 bits long.

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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Analog to Digital Conversion
Analog to Digital Conversion (ADC) is the process of converting continuous analog signals into discrete digital numbers. In this case, the analog voltage signal ranging from 0 V to 5 V is sampled and quantized to fit into a binary format, allowing for digital representation and processing.
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Binary Representation
Binary representation is a way of expressing numbers using only two digits: 0 and 1. In an 8-bit system, numbers can range from 0 to 255, allowing for 256 distinct values. Each bit represents a power of 2, and the combination of these bits encodes the value of the analog signal.
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Quantization
Quantization is the process of mapping a range of values to a finite set of values. In the context of ADC, it involves dividing the analog voltage range into discrete levels. For an 8-bit representation, the 5 V range is divided into 256 levels, with each level corresponding to a specific binary value, which determines how closely the digital representation approximates the original analog signal.
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