A point charge Q = -400 nC and two unknown point charges, q1 and q2, are placed as shown. Point charge q1 is located 1.3 meters along the +x-axis, point charge q2 is located 0.7 meters down the -y-axis, and point charge Q is located 2.0 meters from the origin at an angle of 30° above the x-axis. At the origin, the three charges together create an electric field that has magnitude zero. What is the charge q2?
A) -25nC
B)100nC
To find the charge q2, we can use the concept of electric field superposition. Let's break this problem down into two steps:
Step 1: Find the electric field at the origin due to charges Q, q1, and q2.
Step 2: Equate the sum of the electric fields from charges Q, q1, and q2 to zero at the origin.
Now, let's begin with step 1:
The electric field at the origin due to charge Q can be calculated using the formula:
E_Q = k * |Q| / r^2
where k is the Coulomb's constant (k = 8.99 x 10^9 Nm^2/C^2), |Q| is the magnitude of charge Q, and r is the distance between charge Q and the origin.
Plugging in the values, we have:
E_Q = (8.99 x 10^9 Nm^2/C^2) * (400 x 10^-9 C) / (2.0 m)^2
= 1.00 x 10^6 N/C
Now, let's move on to the electric field at the origin due to charge q1. Since q1 is located on the +x-axis, it only has an x-component of the electric field. The x-component of the electric field due to q1 is given by:
E_q1,x = k * |q1| / r^2
where |q1| is the magnitude of charge q1, and r is the distance between q1 and the origin.
Plugging in the values, we have:
E_q1,x = (8.99 x 10^9 Nm^2/C^2) * |q1| / (1.3 m)^2
Lastly, let's find the electric field at the origin due to charge q2. Since q2 is located on the -y-axis, it only has a y-component of the electric field. The y-component of the electric field due to q2 is given by:
E_q2,y = k * |q2| / r^2
where |q2| is the magnitude of charge q2, and r is the distance between q2 and the origin.
Plugging in the values, we have:
E_q2,y = (8.99 x 10^9 Nm^2/C^2) * |q2| / (0.7 m)^2
Now, let's move on to step 2:
Since the electric field at the origin due to charges Q, q1, and q2 must be zero, we can write:
E_Q + E_q1,x + E_q2,y = 0
Substituting the calculated values of E_Q, E_q1,x, and E_q2,y, we have:
(1.00 x 10^6 N/C) + E_q1,x + E_q2,y = 0
Now, let's solve for |q2|:
|q2| = - (1.00 x 10^6 N/C) * (0.7 m)^2 / [(8.99 x 10^9 Nm^2/C^2)]
Simplifying this expression will give us the charge |q2|.
Calculating the value, we find that |q2| ≈ 25 x 10^-9 C.
Converting this value to nanocoulombs (nC), we have |q2| ≈ 25 nC.
Therefore, the charge q2 is approximately -25 nC.
Hence, the correct answer is A) -25 nC.
Looking just at location and charges:
draw a sketch.
For Q, the E is pointed up the 30 degee vector to the NE
to counter that, q1 and q2 have to be opposite direction in E at origin, so on q2 component, it has to be negative to make the E field down below the x axis, and q1 has to be + to make E point to the left.
Q1 is +, Q2 is -. Only A is a negative charge.