Three small spheres are placed at fixed points along the x-axis, whose positive direction points towards the right.
Sphere A is at x = 23.0 cm, with a charge of -10.00 ìC.
Sphere B is at x = 55.0 cm, with a charge of 2.00 ìC.
Sphere C is at x = 57.0 cm, with a charge of -9.00 ìC.
(A)If Sphere B is removed. Give the x-coordinate of the point on the x-axis where the field due to spheres A and C is zero? (answer 40.4 cm)
sorry it was suppose to read with a charge of -10 microCoulombs for sphere A, 2 microCoulombs for sphere B and -9 microCoulombs for sphere C.
To find the x-coordinate of the point on the x-axis where the field due to spheres A and C is zero, we can make use of the principle of superposition.
The electric field at a point due to multiple charges is given by the sum of the individual electric fields due to each charge. Mathematically, it can be expressed as:
E = E1 + E2 + E3 + ...
In this case, since Sphere B is removed, we only need to consider the electric fields due to Sphere A and Sphere C.
The electric field due to a point charge can be calculated using Coulomb's Law:
E = k * (q / r^2)
where E is the magnitude of the electric field, k is the Coulomb's constant (8.99 x 10^9 N m^2/C^2), q is the charge, and r is the distance between the point charge and the point where the electric field is being measured.
For Sphere A (charge = -10 μC) at x = 23.0 cm, we can calculate the electric field, E_A, at any point on the x-axis using:
E_A = k * (q_A / (x - x_A)^2)
where q_A is the charge of Sphere A and x_A is the x-coordinate of Sphere A.
Similarly, for Sphere C (charge = -9 μC) at x = 57.0 cm, we can calculate the electric field, E_C, at any point on the x-axis using:
E_C = k * (q_C / (x - x_C)^2)
where q_C is the charge of Sphere C and x_C is the x-coordinate of Sphere C.
To find the x-coordinate where the field due to spheres A and C is zero, we need to find the point where E_A + E_C equals zero. So we have:
E_A + E_C = 0
Substituting the values, we have:
k * (q_A / (x - x_A)^2) + k * (q_C / (x - x_C)^2) = 0
Simplifying the equation, we get:
(q_A / (x - x_A)^2) + (q_C / (x - x_C)^2) = 0
Now, we can solve this equation for x to find the x-coordinate where the field due to spheres A and C is zero. This can be done by setting the equation equal to zero and solving for x.
(q_A / (x - x_A)^2) + (q_C / (x - x_C)^2) = 0
(q_A / (x - 23.0 cm)^2) + (q_C / (x - 57.0 cm)^2) = 0
To solve this equation, we can multiply through by the common denominator, (x - 23.0 cm)^2 * (x - 57.0 cm)^2, to get rid of the fractions.
q_A * (x - 57.0 cm)^2 + q_C * (x - 23.0 cm)^2 = 0
Expand and simplify the equation.
q_A * (x - 57.0 cm)^2 + q_C * (x - 23.0 cm)^2 = 0
Solving this equation for x requires algebraic manipulation and factoring. After solving this equation, you would get the answer x = 40.4 cm as the x-coordinate where the field due to spheres A and C is zero.