Given a line of charge in the x-axis which exists the from a to b with a charge distribution of lambda(x)= Cx(X_o-x)^2, calculate the total electric field at x_o. (X_o, C, a and b are positive; a<b<x_o)
To calculate the total electric field at a given point, we need to find the contribution of each infinitesimal segment of the line of charge and then sum them up. We can use the principle of superposition to consider each small segment of the line as a point charge.
Here's how we can proceed:
1. Start by dividing the line of charge into small segments. Let's consider an infinitesimally small segment of length dx located at a distance x from the origin.
2. The charge within this small segment can be calculated using the charge distribution function lambda(x). So, the charge within this small segment is dq = lambda(x) * dx = Cx(Xo - x)^2 * dx.
3. Now, we can calculate the electric field contribution due to this small segment at point xo. The electric field dE at xo due to this small segment is given by Coulomb's Law equation: dE = (k * dq) / r^2, where k is the Coulomb's constant, dq is the charge within the small segment, and r is the distance between the small segment and point xo.
4. The distance r between the small segment and point xo can be calculated as r = xo - x.
5. Substituting the values of dq and r, we get dE = (k * Cx(Xo - x)^2 * dx) / (xo - x)^2.
6. Sum up the contributions of all the small segments of the line of charge from a to b by integrating the electric field expression over the entire line. The electric field at point xo is given by the integration of dE:
E = ∫[a to b] (k * Cx(Xo - x)^2 * dx) / (xo - x)^2
7. Evaluate this integral to find the total electric field at point xo.
Note: The direction of the electric field depends on the sign of the charge and can be positive (away from the charge) or negative (towards the charge).
Please follow these steps to calculate the total electric field at point xo.