Two point charges q1 47C and q2 83 C are fixed along the x

Two point charges (q₁ = -4.7C and q₂ = 8.3 C) are fixed along the x-axis, separated by a distance d = 7.8 cm. Point P is located at (x,y) = (d,d).

1)

What is E_x(P), the value of the x-component of the electric field produced by q₁ and q₂ at point P?N/C

2)

What is E_y(P), the value of the y-component of the electric field produced by q₁ and q₂ at point P?N/C

3)

A third point charge q₃ = 2.3 C is now positioned along the y-axis at a distance d = 7.8 cm from q₁ as shown. What is E_x(P), the x-component of the field produced by all 3 charges at point P?

N/C

4)

Suppose all charges are now doubled (i.e., q₁ = -9.4 C, q₂ = 16.6 C, q₃ = 4.6 C), how will the electric field at P change?

Its magnitude will increase by less than a factor of two and its direction will remain the same

Its magnitude will increase by less than a factor of two and its direction will change

Its magnitude will double and its direction will remain the same

Its magnitude will double and its direction will change

5)

How would you change q₁ (keeping q₂ and q₃ fixed) in order to make the electric field at point P equal to zero?

Increase its magnitude and change its sign

Decrease its magnitude and change its sign

Increase its magnitude and keep its sign the same

Decrease its magnitude and keep its sign the same

There is no change you can make to q₁ that will result in the electric field at point P being equal to zero.

Solution

Q1 = -4.7 uC
Q2 = 8.3 uC
Q3 = 2.3 uC
k = 8.89*10^9
d = 7.8 cm

1)

Ex due to Q2 = 0
Ex due to Q1 = k*Q1 *cos(theta)/(2d^2), where cos(theta) = 1/(sqroot(2))
Ex due to Q1 = -2.4 * 10 ^ 6 N/C

2)

Ey due to Q2 = k * Q2 / d^2 = 1.2 * 10 ^ 7 N/C
Ey due to Q1 = k * Q1 * sin (theta)/((2d)^2) = -2.4 * 10 ^ 6 N/C

3)

Ex due to Q3 = k * Q3 / d^2 = 0.33 * 10 ^ 7 N/C
Net Ex = 9.25 * 10 ^ 5 N/C

4)

If all charges are doubled:
Ex wil become double for Q1, Q2, and Q3
Ey will become double for Q1, Q2 and Q3
Hence Net E will become double with no change in direction as everything is changing proportionally

5)

Consider Ex at P: Ex by Q3 > Ex by Q1
Consider Ey at P: Ey by Q2 > Ey by Q1

Hence Magnitude of charge Q1 should be increased but its sign should not change