The drawing shows a positive point charge +q1, a second point charge q2 that may be positive or negative, and a spot labeled P, all on the same straight line. The distance d between the two charges is the same as the distance between q1 and the spot P. With q2 present, the magnitude of the net electric field at P is twice what it is when q1 is present alone. Given that q1 = +2.86 µC, determine the magnitude |q2| when q2 is the following.

Question

The drawing shows a positive point charge +q1, a second point charge q2 that may be positive or negative, and a spot labeled P, all on the same straight line. The distance d between the two charges is the same as the distance between q1 and the spot P. With q2 present, the magnitude of the net electric field at P is twice what it is when q1 is present alone. Given that q1 = +2.86 µC, determine the magnitude |q2| when q2 is the following.

Answer 1

To determine the magnitude of q2, we can use the concept of electric field and superposition principle.

The electric field produced by a point charge q at a distance r from it is given by the equation:

E = k * (|q| / r^2)

Where E is the electric field, k is Coulomb's constant (approximately 8.99 x 10^9 Nm^2/C^2), q is the charge, and r is the distance from the charge.

In this case, we have two charges, q1 and q2, and we are comparing the magnitude of the net electric field at point P when only q1 is present and when both q1 and q2 are present.

Let's denote the magnitude of the net electric field at P with only q1 as EP1, and the magnitude of the net electric field at P with both q1 and q2 as EP2.

According to the problem statement, EP2 is twice EP1. Mathematically, we can write this as:

EP2 = 2 * EP1

Now, let's substitute the expressions for electric field:

k * (|q1| / d^2) = 2 * k * ((|q1| + |q2|) / d^2)

Simplifying and rearranging the equation, we get:

|q1| = 2 * (|q1| + |q2|)

Dividing both sides by 2, we have:

|q1| / 2 = |q1| + |q2|

Subtracting |q1| from both sides, we get:

|q1| / 2 - |q1| = |q2|

Simplifying further, we have:

-|q1| / 2 = |q2|

Now, we can substitute the given value for |q1|, which is +2.86 µC (microCoulombs) or +2.86 x 10^-6 C (Coulombs), into the equation:

-2.86 x 10^-6 C / 2 = |q2|

Simplifying, we find:

|q2| = -1.43 x 10^-6 C

Therefore, the magnitude of |q2| is 1.43 µC (microCoulombs) or 1.43 x 10^-6 C (Coulombs), and its sign is negative.

Note: The negative sign indicates that q2 is negative in this case.

Answer 2

To find the magnitude of |q2| when the net electric field at P is twice what it is when q1 is present alone, we can use the concept of electric field superposition.

Step 1: Express the electric field due to q1 alone
The electric field at spot P due to q1 alone is given by Coulomb's law:
E1 = k * |q1| / r^2

Step 2: Express the electric field due to q2 alone
The electric field at spot P due to q2 alone is given by Coulomb's law:
E2 = k * |q2| / r^2

Step 3: Express the net electric field when both charges are present
The net electric field at spot P when both q1 and q2 are present, is given by:
E_net = E1 + E2

Step 4: Use the given information to set up the equation
From the problem statement, it is given that the magnitude of the net electric field at P is twice what it is when q1 is present alone. Mathematically, this can be written as:
E_net = 2 * E1

Substituting the expressions for E_net and E1, we get:
k * |q1| / r^2 + k * |q2| / r^2 = 2 * (k * |q1| / r^2)

Step 5: Cancel out common terms and rearrange the equation
Cancelling out common terms, we have:
|q1| + |q2| = 2 * |q1|

Rearranging, we get:
|q2| = 2 * |q1| - |q1|
|q2| = |q1|

Step 6: Substitute the given value of q1
The given value of q1 is +2.86 µC, which can be written as 2.86 * 10^-6 C.

Therefore, |q2| = |q1| = 2.86 * 10^-6 C.

Thus, the magnitude of |q2| is also 2.86 µC or 2.86 * 10^-6 C.

Answer 3

E=kq/r^2

so if r=P + P for the farther charge

then q2 must be four times as large as q1, as the distance is twice as far.