Two positive point charges Q and Q’ having charges of +4 μC and +1 μC respectively are fixed at a distance 2 m apart in vacuum.A third point charge q having a charge +0.2 μC is to be placed between Q and Q’, at a distance x from Q’. If q remains at rest, what is the value of distance x?
Coulomb's
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+4*10^-6 C at x = 2
+1*10^-6 C at x =0 note locations reversed so x is distance from Q'
it does not matter what the value of Q between is, we must find where E = 0 so call it Q
k *1*Q /x^2 = k * 4 * Q/ (2-x)^2
4 x^2 = (2-x)^2
4 x^2 = 4 - 4 x + x^2
3 x^2 + 4 x - 4 = 0
(3x-2)(x+2) = 0
positive x = 2/3 meter = distance from smaller charge
I know I need to use columns law to do this but I don’t know how to proceed.
To find the value of distance x, we can use Coulomb's Law, which states that the force between two point charges is directly proportional to the product of their charges and inversely proportional to the square of the distance between them.
The force exerted by the charges Q and q on each other is given by:
F1 = k * (Q * q) / r^2,
where k is the Coulomb's constant (k ≈ 9 × 10^9 N m^2/C^2), Q is the charge of Q (4 μC = 4 × 10^-6 C), q is the charge of q (0.2 μC = 0.2 × 10^-6 C), and r is the distance between Q and q (x m).
Similarly, the force exerted by the charges Q' and q on each other is given by:
F2 = k * (Q' * q) / (2 + x)^2.
Since the system is at equilibrium (q remains at rest), the two forces must balance each other out. Thus, F1 = F2.
Substituting the given values, we have:
k * (Q * q) / r^2 = k * (Q' * q) / (2 + x)^2.
Canceling out the common terms, we get:
(Q * q) / r^2 = (Q' * q) / (2 + x)^2.
Plugging in the values, we have:
(4 × 10^-6 C * 0.2 × 10^-6 C) / (x^2) = (1 × 10^-6 C * 0.2 × 10^-6 C) / (2 + x)^2.
Simplifying further, we get:
8 × 10^-12 / (x^2) = 0.2 × 10^-12 / (2 + x)^2.
Now, cross-multiplying, we have:
8 × 10^-12 * (2 + x)^2 = 0.2 × 10^-12 * (x^2).
Expanding both sides of the equation, we get:
8 × 10^-12 * (4 + 4x + x^2) = 0.2 × 10^-12 * (x^2).
Simplifying further:
32 + 32x + 8x^2 = 0.2x^2.
Rearranging the equation, we have:
32x + 8x^2 - 0.2x^2 - 32 = 0.
Combining like terms, we get:
7.8x^2 + 32x - 32 = 0.
This quadratic equation can be solved using various methods, such as factoring, completing the square, or using the quadratic formula.
By solving the equation, we find that the value of x is approximately -4.99 meters or 0.425 meters. Since distance cannot be negative, the value of x is approximately 0.425 meters.
Therefore, the value of distance x is approximately 0.425 meters.