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Chapter 25: Q. 73 (page 713)

The wire in FIGURE P25.73 has linear charge density $λ$ . What is the electric potential at the center of the semicircle?

Short Answer

Expert verified

The electric potential at the center of the semicircle is $V = k λ (2 I n (3) + π)$ .

Step by step solution

01

Given Information

We need to find the electric potential at the center of the semicircle.

02

Simplify

The potential at the point of interest will be the sum of potentials coming from the straight lines and the curved section. By symmetry, it is clear that the potential coming from the two straight segments are equal. So, we can calculate one of them, and the potential coming from both will be twice in our result.

Therefore, we can write

$V = k (2 \int_{0}^{2 R} \frac{λ d z}{x + R} + \int_{0}^{πR} \frac{λ d x}{R})$

Factorizing the linear charge density, we can simplify

$V = k λ (\int_{0}^{2 R} \frac{d x}{x + R} + \frac{1}{R} \int_{0}^{πR} d x)$

The result will be

$k λ (2 [I n |3 R| - I n |R|] + π) = k λ (2 I n (3) + π)$

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Most popular questions from this chapter

Show that $1 V / m = 1 N / C$ .

Physicists often use a different unit of energy, the electron volt, when dealing with energies at the atomic level. One electron volt, abbreviated eV, is defined as the amount of kinetic energy gained by an electron upon accelerating through a 1.0 V potential difference. a. What is 1 electron volt in joules? b. What is the speed of a proton with 5000 eV of kinetic energy?

One form of nuclear radiation, beta decay, occurs when a neutron changes into a proton, an electron, and a neutral particle called a neutrino: $n \to p^{+} + e^{-} + v$ where $n$ is the symbol for a neutrino. When this change happens to a neutron within the nucleus of an atom, the proton remains behind in the nucleus while the electron and neutrino are ejected from the nucleus. The ejected electron is called a beta particle. One nucleus that exhibits beta decay is the isotope of hydrogen $3 H$ , called tritium, whose nucleus consists of one proton (making it hydrogen) and two neutrons (giving tritium an atomic mass $m = 3 u$ ). Tritium is radioactive, and it decays to helium: ${}^{3}H \to {}^{3}H e + e^{-} + n$

a. Is charge conserved in the beta decay process? Explain.

b. Why is the final product a helium atom? Explain.

c. The nuclei of both ${}^{3}H and {}^{3}H e$ have radii of $1.5 \times 10^{- 15} m$ . With what minimum speed must the electron be ejected if it is to escape from the nucleus and not fall back?

Two point charges 2.0 cm apart have an electric potential energy -180 mJ. The total charge is 30 nC. What are the two charges?

The electric field strength is $20, 000 N / C$ inside a parallel-plate capacitor with a $1.0 m m$ spacing. An electron is released from rest at the negative plate. What is the electron’s speed when it reaches the positive plate?

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