A charge of 1.5uC is placed at the origin, and a charge of 3uC is placed at x = 1.5m. Locate the point between the two charges where the electric field is zero?
Q1 P Q2
x------------*-----------.
0
<---------1.5m----------->
the distance to the left charge is x, the distance to the right charge is (1.5-x)
E=0=kQ1/x^2-kQ2/(1.5-x)^2
the second term is minus because E at the test point is in the opposite direction to the first E.
solve for x
To locate the point between the two charges where the electric field is zero, we need to consider the equation for the electric field due to a point charge:
E = k * Q / r^2
where E is the electric field, k is the electrostatic constant (9 x 10^9 Nm^2/C^2), Q is the charge, and r is the distance from the charge.
Let's denote the distance from the origin to the unknown point as "x". The electric field due to charge Q1 at the origin can be defined as:
E1 = k * 1.5uC / (x)^2
The electric field due to charge Q2 at 1.5m can be defined as:
E2 = k * 3uC / (1.5 - x)^2
Since we want the point where the electric field is zero, E1 = -E2. Thus, we can set up the following equation:
k * 1.5uC / x^2 = -k * 3uC / (1.5 - x)^2
Now we can solve for x:
1.5 / x^2 = -3 / (1.5 - x)^2
Multiplying both sides by (x^2)(1.5 - x)^2:
1.5(1.5 - x)^2 = -3x^2
Expanding and rearranging:
2.25 - 3x + 1.5x^2 = -3x^2
3x^2 - 3x - 2.25 = 0
Dividing by 3:
x^2 - x - 0.75 = 0
This equation is a quadratic equation that can be factored or solved using the quadratic formula. The factors of the equation are (x - 1.5)(x + 0.5) = 0.
Setting each factor equal to zero:
x - 1.5 = 0 or x + 0.5 = 0
Therefore, x = 1.5 or x = -0.5.
However, since the distance cannot be negative in this case, the only valid solution is x = 1.5. Thus, the point between the two charges where the electric field is zero is at x = 1.5m.
To locate the point between the two charges where the electric field is zero, we can use the principle of superposition. The electric field at any point is the vector sum of the electric fields due to each individual charge.
First, we need to find the electric field at the point between the two charges, due to each charge separately.
Let's call the point where we want to find the electric field E_Total. The electric field due to charge Q1 at point E_Q1 can be calculated using Coulomb's Law:
E_Q1 = k * Q1 / r^2
where k is the Coulomb's constant (9 × 10^9 N·m^2/C^2), Q1 is the charge (1.5 μC), and r is the distance between the point and charge Q1.
Similarly, for charge Q2, the electric field at point E_Q2 can be calculated using the same formula:
E_Q2 = k * Q2 / r^2
where Q2 is the charge (3 μC), and r is the distance between the point and charge Q2.
Given that the charges are placed at the origin (Q1) and x = 1.5 m (Q2), the distances will be r_1 = x and r_2 = x - 1.5, respectively.
Now, using the principle of superposition, the total electric field at the point E_Total is the vector sum of E_Q1 and E_Q2:
E_Total = E_Q1 + E_Q2
To find the point where the electric field is zero, we need to set E_Total to zero and solve for x:
E_Total = E_Q1 + E_Q2 = 0
Substituting the expressions for E_Q1 and E_Q2:
k * Q1 / r_1^2 + k * Q2 / r_2^2 = 0
Plugging in the values:
(9 × 10^9 N·m^2/C^2) * (1.5 μC) / x^2 + (9 × 10^9 N·m^2/C^2) * (3 μC) / (x - 1.5)^2 = 0
Simplifying and solving this equation will give us the value of x where the electric field is zero.