Two point charges, 4.0×10-6 C and -1.0×10-6 C, are located on the x axis at = -1.0 cm and = 3.0 cm.
(a) Determine the electric field at the origin.
(b)Determine the x coordinate of a point on the positive x axis where the electric field is zero; i.e., a test charge placed at this point would experience no force.
To determine the electric field at the origin, we need to use the principle of superposition, which states that the electric field at a point due to multiple point charges is the vector sum of the electric fields created by each individual point charge.
(a) Electric Field at the Origin:
Let's denote Q1 as the charge with magnitude 4.0×10^(-6) C located at -1.0 cm on the x-axis, and Q2 as the charge with magnitude -1.0×10^(-6) C located at 3.0 cm on the x-axis.
The electric field created by a point charge at a given point is given by Coulomb's Law:
E = k * (Q / r^2)
Where E is the electric field, k is the Coulomb's constant (approximated as 9.0×10^9 Nm^2/C^2), Q is the charge magnitude, and r is the distance from the charge to the point at which the electric field is measured.
The electric field created by Q1 at the origin is:
E1 = k * (Q1 / r1^2)
where r1 is the distance between Q1 and the origin, which is the absolute value of -1.0 cm.
The electric field created by Q2 at the origin is:
E2 = k * (Q2 / r2^2)
where r2 is the distance between Q2 and the origin, which is the absolute value of 3.0 cm.
To find the net electric field at the origin, we need to calculate the vector sum of E1 and E2:
E_net = E1 + E2
Substituting the given values:
E1 = (9.0×10^9 Nm^2/C^2) * (4.0×10^(-6) C) / (0.01 m)^2
E2 = (9.0×10^9 Nm^2/C^2) * (-1.0×10^(-6) C) / (0.03 m)^2
E_net = E1 + E2
Calculate E_net using the given values.
(b) X Coordinate with Zero Electric Field:
To find the x-coordinate where the electric field is zero on the positive x-axis, we need to find a point where the net electric field is zero. Let's assume this x-coordinate is x.
At this point, the electric field created by Q1 at x is equal to the electric field created by Q2 at x:
E1_x = E2_x
Set up the equation using the known formulas for the electric field:
k * (Q1 / (x + 0.01 m)^2) = k * (Q2 / (3 m - x)^2)
Solve this equation to find the value of x.
Note: When calculating values, be cautious of units. Convert the given distances to meters for consistency in the calculations.