Two particles are fixed to an x axis: particle 1 of charge q1 = 1.73 ¡Ñ 10-8 C at x = 30.0 cm and particle 2 of charge q2 = -4.41q1 at x = 78.0 cm. At what coordinate on the x axis is the electric field produced by the particles equal to zero?

distances from charges

x1 = x-30
x2 = 78-x

q1/x1^2 = q2/x2^2

1/x1^2 = 4.41 /x2^2

1/(x-30)^2 = 4.41/(78-x)^2

6084 - 156 x + x^2 = 4.41(x^2 -60 x + 900)

solve the quadratic

To find the coordinate on the x-axis where the electric field produced by the particles is zero, we can use the principle of superposition.

The electric field at a point on the x-axis due to particle 1 can be given by the following equation:

E1 = k * q1 / r1^2

where E1 is the electric field, k is Coulomb's constant (k = 8.99 * 10^9 N m^2/C^2), q1 is the charge of particle 1, and r1 is the distance between particle 1 and the point on the x-axis.

Similarly, the electric field at the same point on the x-axis due to particle 2 can be given by:

E2 = k * q2 / r2^2

where E2 is the electric field, q2 is the charge of particle 2, and r2 is the distance between particle 2 and the point on the x-axis.

Since we want the total electric field to be zero, we can set E1 + E2 = 0.

Substituting the values, we get:

k * q1 / r1^2 + k * q2 / r2^2 = 0

Given that q2 = -4.41 * q1, we can substitute it in the equation:

k * q1 / r1^2 - 4.41 * k * q1 / r2^2 = 0

Now we have an equation with two variables, r1 and r2. We need to solve this equation to find the values of r1 and r2.

Using the given distances, r1 = 30 cm = 0.3 m and r2 = 78 cm = 0.78 m, we can substitute these values in the equation:

(8.99 * 10^9 N m^2/C^2) * q1 / (0.3 m)^2 - 4.41 * (8.99 * 10^9 N m^2/C^2) * q1 / (0.78 m)^2 = 0

Simplifying the equation, we get:

9.194 * 10^18 q1 - 2.126 * 10^18 q1 = 0

Solving for q1, we find:

7.068 * 10^18 q1 = 0

So, q1 = 0.

This means that for the electric field to be zero at any point on the x-axis, the charge of particle 1 (q1) must be zero.

Therefore, at any coordinate on the x-axis, the electric field produced by the particles will be equal to zero.