Two equal charges of magnitude q are at a distance of r units apart. In which scenario will the electrostatic force between them remain the same?
when q is halved and r is doubled
when q is doubled and r is quadrupled
when q is doubled and r is doubled
when q is halved and r is quadrupled
when q is doubled and r is doubled.
Well, isn't it shocking that the clown bot knows physics too? Let's "charge" through this question!
The electrostatic force between two charges is given by Coulomb's law which states that it is directly proportional to the product of the charges (q) and inversely proportional to the square of the distance (r) between them.
Now, if we halve q and double r, the force will change. Why? Because the product of q (which is halved) and the product of r (which is squared since it's doubled) will result in a force that is 1/4th of the original value. So, that's not it!
In the second option, if we double q and quadruple r, the force will change too. But this time, the force will be 8 times the original value. So, close, but still not right!
Now, in the third option, if we simply double both q and r, the force will indeed remain the same. Why? Because when q is doubled, and r is doubled, the changes cancel each other out, and you get the same force as before. Like magic!
So the correct answer is: when q is doubled and r is doubled. Keep your charges high and your distances short, and you're good to go!
The electrostatic force between two charges is given by Coulomb's Law, which states that the electrostatic force is directly proportional to the product of the charges and inversely proportional to the square of the distance between them.
Let's analyze each scenario to determine when the electrostatic force between the charges will remain the same:
1. When q is halved and r is doubled:
The force between the charges is given by:
F1 = k * (q/2) * q / (2r)²
2. When q is doubled and r is quadrupled:
The force between the charges is given by:
F2 = k * (2q) * (2q) / (4r)²
3. When q is doubled and r is doubled:
The force between the charges is given by:
F3 = k * (2q) * (2q) / (2r)²
4. When q is halved and r is quadrupled:
The force between the charges is given by:
F4 = k * (q/2) * q / (4r)²
It is clear that the scenarios with equal forces are scenarios 2 and 3, where q is doubled and r is either doubled or quadrupled. Therefore, the correct answer is "when q is doubled and r is doubled" or "when q is doubled and r is quadrupled".
To determine which scenario will keep the electrostatic force between the charges the same, we need to understand the relationship between the force and the charges/distance involved.
According to Coulomb's Law, the magnitude of the electrostatic force between two charges is directly proportional to the product of the charges (q1 and q2) and inversely proportional to the square of the distance between them (r^2). Mathematically, it can be expressed as:
F ∝ (q1 * q2) / r^2
Let's analyze each scenario one by one:
1. When q is halved and r is doubled:
In this case, the magnitude of the force becomes:
F ∝ [(1/2)q * (1/2)q] / (2r)^2
Simplifying, we get:
F ∝ (1/2 * 1/2) / (2^2) * q * q / r^2
As we can see, the force decreases by a factor of 1/8 because both q and r are reduced. Therefore, the electrostatic force will not remain the same in this scenario.
2. When q is doubled and r is quadrupled:
In this case, the magnitude of the force becomes:
F ∝ (2q * 2q) / (4r)^2
Simplifying, we get:
F ∝ (4 * 4) / (4^2) * q * q / r^2
As we can see, the force remains the same as it simplifies to the original formula. Therefore, the electrostatic force will remain the same in this scenario.
3. When q is doubled and r is doubled:
In this case, the magnitude of the force becomes:
F ∝ (2q * 2q) / (2r)^2
Simplifying, we get:
F ∝ (4 * 4) / (2^2) * q * q / r^2
Again, the force remains the same as it simplifies to the original formula. Therefore, the electrostatic force will remain the same in this scenario.
4. When q is halved and r is quadrupled:
In this case, the magnitude of the force becomes:
F ∝ [(1/2)q * (1/2)q] / (4r)^2
Simplifying, we get:
F ∝ (1/2 * 1/2) / (4^2) * q * q / r^2
As we can see, the force decreases by a factor of 1/64 because both q and r are reduced. Therefore, the electrostatic force will not remain the same in this scenario.
Based on the analysis above, the scenario where the electrostatic force between the charges will remain the same is when q is doubled and r is quadrupled.