The electrical charge distribution on a circular plate of radius R meters is given by the function f(x,y) = kx (1- sin y) in coulomb/m^2. Integrate the function over the entire plate to find the total charge Q.
To find the total charge Q on the circular plate, we need to integrate the charge density function f(x, y) over the entire plate.
The charge density function is given by f(x, y) = kx(1 - sin y).
To integrate this function over the entire plate, we need to define the limits of integration. Since the plate is circular and has a radius R, we can use polar coordinates to describe the plate. Let's use r as the radial coordinate and θ as the angular coordinate.
In polar coordinates, x = rcosθ and y = rsinθ.
The charge density function in terms of polar coordinates becomes f(r, θ) = krcosθ(1 - sinθ).
To integrate this function over the entire plate, we integrate with respect to r from 0 to R, and with respect to θ from 0 to 2π (a full revolution around the circle).
So, the integral becomes:
Q = ∫∫(0 to 2π)(0 to R) kr*cosθ(1 - sinθ) rdrdθ
We can evaluate this integral step by step. First, let's integrate with respect to r:
∫(0 to R) kr*cosθ(1 - sinθ) dr = k*cosθ(1 - sinθ) ∫(0 to R) r dr
Applying the limits of integration, we get:
Q = k*cosθ(1 - sinθ) * ½R²
Now, let's integrate with respect to θ:
∫(0 to 2π) k*cosθ(1 - sinθ) * ½R² dθ = k*½R² [-sinθ + ½cosθ] (0 to 2π)
Evaluating the limits, we get:
Q = kR² (0 - ½) = -½kR²
So, the total charge Q on the circular plate is -½kR² coulombs.
(Note: The negative sign indicates that the charge on the plate is negative or opposite in sign to the conventionally defined positive charge.)