Charge is distributed uniformly along the x axis with density ßx and uniformly along the y axis with density ßy. Calculate the electric field at the points
(a) r = a i + b j, and
(b) r = c k.
To calculate the electric field at a given point, we can use Coulomb's Law. Coulomb's Law states that the electric field at a point P due to a charged particle is proportional to the magnitude of the charge and inversely proportional to the square of the distance between the point and the particle. We can then sum up the electric fields due to each charged particle to get the total electric field at a point.
In this case, we have charge distributions along the x-axis and the y-axis. Let's start by calculating the electric field at point P = a i + b j.
(a) Electric field at r = a i + b j:
To calculate the electric field at this location, we need to consider the contribution of the charges distributed along the x-axis and the y-axis separately.
Electric field due to charges distributed along the x-axis:
Along the x-axis, the charge density is ßx. We can consider small charge elements, dq, and calculate the electric field from each of these infinitesimal charges. Since the distribution is uniform, we can assume that the charge density is constant over each small charge element.
Let's consider a small charge element located at x = xi, where xi is a small distance along the x-axis. The charge in this small element, dq, is given by dq = ßx * dx, where dx is the width of the small charge element.
The electric field due to this charge element at the point P = a i + b j is given by Coulomb's Law as:
dE_x = k * (dq / r_x^2) * (i / r_x),
where k is the Coulomb's constant, dq is the charge element, r_x is the distance between the charge element and the point P projected onto the x-axis, and i is the unit vector in the x-direction.
Since the charge density is constant, we can write dq = ßx * dx. The distance r_x is given by r_x = (a - xi). Note that a - xi represents the distance between the charge element at x = xi and the point P projected onto the x-axis.
Integrating over the entire length of the x-axis, the total electric field contribution from the charges distributed along the x-axis is:
E_x = ∫[x=0 to x=L] dE_x,
where L is the length of the x-axis.
Similarly, we need to consider the electric field due to charges distributed along the y-axis.
Electric field due to charges distributed along the y-axis:
Along the y-axis, the charge density is ßy. Again, we can assume that the charge density is constant over each small charge element.
Let's consider a small charge element located at y = yj, where yj is a small distance along the y-axis. The charge in this small element, dq, is given by dq = ßy * dy, where dy is the width of the small charge element.
The electric field due to this charge element at the point P = a i + b j is given by Coulomb's Law as:
dE_y = k * (dq / r_y^2) * (j / r_y),
where dq is the charge element, r_y is the distance between the charge element and the point P projected onto the y-axis, and j is the unit vector in the y-direction.
Since the charge density is constant, we can write dq = ßy * dy. The distance r_y is given by r_y = (b - yj). Note that b - yj represents the distance between the charge element at y = yj and the point P projected onto the y-axis.
Integrating over the entire length of the y-axis, the total electric field contribution from the charges distributed along the y-axis is:
E_y = ∫[y=0 to y=H] dE_y,
where H is the height of the y-axis.
Finally, we can calculate the total electric field at point P = a i + b j by summing up the electric field contributions along the x-axis and the y-axis:
E = E_x + E_y.
(b) Electric field at r = c k:
Similarly, to calculate the electric field at this location, we need to consider the contribution of the charges distributed along the x-axis and the y-axis separately.
Since the point P = c k lies along the z-axis, the electric field contribution from the charges distributed along the x-axis will be zero since the x-coordinate of the charge elements is zero. Thus, we only need to consider the electric field due to charges distributed along the y-axis.
By following the same steps as in part (a) for charges distributed along the y-axis, we can calculate the electric field at the point P = c k as:
E = E_y,
where E_y is the total electric field contribution from the charges distributed along the y-axis.