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1. Intro to Physics Units1h 29m
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22. The First Law of Thermodynamics1h 26m
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24. Electric Force & Field; Gauss' Law3h 42m
25. Electric Potential1h 51m
26. Capacitors & Dielectrics2h 2m
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35. Special Relativity2h 10m

25. Electric Potential

Electric Potential

Physics

25. Electric Potential

Electric Potential

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Problem 74

Textbook Question

The wire in FIGURE P25.74 has linear charge density λ. What is the electric potential at the center of the semicircle?

Verified step by step guidance

Step 1: Understand the problem. The wire consists of a semicircular arc and two straight segments. The linear charge density of the wire is λ, and we need to calculate the electric potential at the center of the semicircle (denoted as the black dot in the figure). The electric potential is a scalar quantity, and contributions from different parts of the wire can be added directly.

Step 2: Recall the formula for electric potential due to a charged element. The potential at a point due to a small charge element dq is given by V = k_e * dq / r, where k_e is Coulomb's constant, dq is the charge of the element, and r is the distance from the charge element to the point of interest.

Step 3: Analyze the semicircular arc. The arc has radius R, and every point on the arc is equidistant from the center. The total charge on the arc is λ * (πR), where πR is the length of the semicircular arc. The potential contribution from the arc is V_arc = k_e * (λ * πR) / R.

Step 4: Analyze the straight segments. Each straight segment has a length of 2R. The distance from any point on the straight segment to the center varies, so we need to integrate to find the potential contribution. For a small element dx on the straight segment, dq = λ * dx, and the distance to the center is √(x^2 + R^2). The potential contribution from one segment is V_segment = ∫(k_e * λ * dx / √(x^2 + R^2)) from x = -2R to x = 0 for the left segment, and similarly for the right segment from x = 0 to x = 2R.

Step 5: Combine the contributions. The total electric potential at the center is the sum of the contributions from the semicircular arc and the two straight segments: V_total = V_arc + 2 * V_segment. Perform the integration for the straight segments and add the results to the potential from the arc to find the final expression for the electric potential.

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Key Concepts

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

Electric Potential

Electric potential, often denoted as V, is the amount of electric potential energy per unit charge at a point in an electric field. It is a scalar quantity measured in volts (V) and indicates the work done to move a unit positive charge from a reference point to a specific point in the field without any acceleration.

Linear Charge Density

Linear charge density, represented by the symbol λ (lambda), is defined as the amount of electric charge per unit length along a line. It is expressed in coulombs per meter (C/m) and is crucial for calculating the electric field and potential generated by charged objects, such as the wire in the semicircular configuration.

Integration in Electric Potential Calculation

To find the electric potential due to a continuous charge distribution, integration is often used. This involves summing the contributions to the potential from infinitesimal charge elements along the charged object. In this case, the semicircular wire's potential at the center requires integrating the contributions from each segment of the wire, taking into account the geometry and charge density.

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