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27. Resistors & DC Circuits

Intro to Current

Physics

27. Resistors & DC Circuits

Intro to Current

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3:34 minutes

Problem 96

Textbook Question

The LHC accelerator at CERN (Switzerland) is designed to carry a 0.58-A beam of protons ( q = 1.6 x 10^-19 C ) traveling at very nearly the speed of light ( 3.0 x 10⁸ m/s) around a ring 27 km in circumference. How many protons are in the beam?

Verified step by step guidance

Step 1: Understand the problem. We are tasked with finding the number of protons in the beam. To do this, we need to relate the current (I), the charge of a proton (q), and the speed of the protons (v) to the number of protons in the beam.

Step 2: Recall the relationship between current and charge. Current (I) is defined as the rate of flow of charge, which can be expressed as:

I = \frac{q \cdot N}{t}

, where q is the charge of a single proton, N is the number of protons passing a point, and t is the time.

Step 3: Express the time (t) in terms of the distance traveled and the speed of the protons. Since the protons are traveling at nearly the speed of light (v = 3.0 x 10⁸ m/s) and the circumference of the ring is 27 km (27,000 m), the time for one complete revolution is:

t = \frac{\text{distance}}{\text{speed}} = \frac{27,000}{3.0 \times 10^8}

Step 4: Rearrange the current equation to solve for the number of protons (N). Substituting t into the equation, we get:

N = \frac{I \cdot t}{q}

. Replace t with the expression derived in Step 3:

N = \frac{I \cdot \frac{27,000}{3.0 \times 10^8}}{q}

Step 5: Substitute the known values into the equation. Use I = 0.58 A, q = 1.6 x 10⁻¹⁹ C, and simplify the expression to find the number of protons in the beam. Ensure units are consistent throughout the calculation.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Current and Charge

Current (I) is the flow of electric charge, measured in amperes (A). In this context, a 0.58-A beam of protons indicates that 0.58 coulombs of charge pass a point in the beam every second. The charge of a single proton is approximately 1.6 x 10⁻¹⁹ coulombs, which allows us to relate the total current to the number of protons in the beam.

Proton Count Calculation

To find the number of protons in the beam, we can use the relationship between current, charge, and time. The total charge (Q) in the beam over a specific time (t) can be calculated using the formula Q = I × t. By dividing the total charge by the charge of a single proton, we can determine the total number of protons present in the beam.

Speed of Light and Relativistic Effects

The protons in the LHC travel at speeds very close to the speed of light, which introduces relativistic effects. While this question primarily focuses on the number of protons, understanding that they are moving at relativistic speeds is important for comprehending their behavior in the accelerator. At such speeds, the mass and energy of the protons increase, affecting their dynamics in the accelerator.

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Textbook Question

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Textbook Question

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