- #1
ewr
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In the problem attached, why is the integral become just the length of line cf?
Last edited:
Calculating Length of Line cf: A Math Problem Explained
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SUMMARY
The discussion focuses on the calculation of the length of line cf using the concept of line integrals. It establishes that the integral simplifies to the length of the line due to the definition of a line integral, specifically when the electric field vector is aligned with the path of integration. The mathematical expression provided illustrates that the integral of the electric field over the distance d results in the negative product of the electric field and the length of the line.
PREREQUISITES- Understanding of line integrals in vector calculus
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- Study the definition and properties of line integrals in vector calculus
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Students and professionals in physics, mathematics, and engineering who are looking to deepen their understanding of line integrals and their applications in calculating physical quantities like work and distance.
In the problem attached, why is the integral become just the length of line cf?
- #2
Nugatory
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ewr said:
That result pretty much follows from the definition of a line integral. If you're not familiar with that concept, google will find you some pretty decent explanations.
- #3
jtbell
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Note that in part (a) the integral becomes just the length of line if, that is, d.
$$V_f - V_i = -\int^f_i {\vec E \cdot d\vec s}
= -\int^f_i {E(\cos 0^\circ)ds} = -E \int^f_i {ds} = -Ed$$
$$V_f - V_i = -\int^f_i {\vec E \cdot d\vec s}
= -\int^f_i {E(\cos 0^\circ)ds} = -E \int^f_i {ds} = -Ed$$
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