An electric field varies in space according to the equation E(x) = axe−bx𝐸(𝑥) = 𝑎𝑥𝑒−𝑏𝑥 , where a = 2.395 N(Cm)−1𝑎 = 2.395 N(Cm)−1 and b = 0.491 m−1𝑏 = 0.491 m−1 . (Include a minus sign if required.) (a) For what value of x does the electric field have its largest value, xmax𝑥max ?
Question
An electric field varies in space according to the equation E(x) = axe−bx𝐸(𝑥) = 𝑎𝑥𝑒−𝑏𝑥 , where a = 2.395 N(Cm)−1𝑎 = 2.395 N(Cm)−1 and b = 0.491 m−1𝑏 = 0.491 m−1 . (Include a minus sign if required.) (a) For what value of x does the electric field have its largest value, xmax𝑥max ?
Solution
To find the maximum value of the electric field, we need to take the derivative of the electric field function with respect to x and set it equal to zero. This is because the maximum or minimum of a function occurs where its derivative is zero.
The electric field function is given by E(x) = axe^(-bx).
Let's take the derivative of E(x) with respect to x:
dE/dx = a * e^(-bx) - a * b * x * e^(-bx)
Setting this equal to zero gives:
a * e^(-bx) - a * b * x * e^(-bx) = 0
We can factor out a * e^(-bx) to simplify:
a * e^(-bx) * (1 - b * x) = 0
Setting this equal to zero gives two solutions: x = 0 and x = 1/b.
However, x = 0 is a minimum of the function, not a maximum. So, the maximum occurs at x = 1/b.
Given that b = 0.491 m^-1, the maximum electric field occurs at x = 1/b = 1/0.491 m^-1 = 2.035 m.
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