A positive charge Q is on the y-axis at a distance a from the origin, and another positive charge q is on the x-axis at a distance b from the origin.
A) For what values(s) of b is the x-component of the force on q a minimum?
B) For what values(s) of b is the x-component of the force on q a maximum?
A) For what values(s) of b is the x-component of the force on q a minimum?
Hmm, let me think... You know, forces can be tricky little buggers. But fear not, for I have an answer that will blow you away! The x-component of the force on q will be minimum when the charges Q and q are aligned along the x-axis. So, for any value of b where charge q is directly in line with charge Q, the x-component of the force on q will be at its minimum - just like my motivation on Monday mornings.
B) For what values(s) of b is the x-component of the force on q a maximum?
Ah, the quest for the maximum force! Now we're talking! The x-component of the force will be maximum when charge q is perpendicular to charge Q. Picture it - charge q sitting on the x-axis, giving the force its all like a weightlifter competing in the charge Olympics. So, for any value of b where charge q is at a 90-degree angle to charge Q, the x-component of the force on q will reach its maximum - just like the volume of my laughter when I eat way too much ice cream!
To determine the minimum and maximum values of the x-component of the force on the charge q, we need to consider the force due to the charge Q acting on q.
The force between two charges is given by Coulomb's Law:
F = k * ((Q * q) / r^2)
Where:
- F is the force between the charges
- k is the electrostatic constant (k = 9 × 10^9 N m^2 / C^2)
- Q and q are the magnitudes of the charges
- r is the distance between the charges
Now let's analyze the x-component of the force on q:
A) For the x-component of the force on q to be a minimum:
We need to find the values of b where the x-component of the force is the smallest. Since the charges are positive, the x-component of the force will be attractive (towards the charge Q).
The x-component of the force is given by:
Fx = F * cos(theta)
Where theta is the angle between the x-axis and the direction of the force. In this case, theta = 90 degrees.
So, Fx = F * cos(90) = 0
This means that the x-component of the force is zero. Hence, there is no minimum value for the x-component of the force.
B) For the x-component of the force on q to be a maximum:
We need to find the values of b where the x-component of the force is the largest. Since the charges are positive, the x-component of the force will be attractive (towards the charge Q).
The maximum value occurs when the charges are aligned along the x-axis. In other words, if the charge Q is placed to the right of charge q, such that b = a.
The x-component of the force is given by:
Fx = F * cos(theta)
Since theta = 0 degrees (charges are aligned along the x-axis), cos(theta) = 1.
So, Fx = F * cos(0) = F
This means that the x-component of the force is equal to the total force between the charges. Hence, the maximum value for the x-component of the force occurs when b = a.
Therefore, for part B, the x-component of the force on q is maximum when b = a.
To find the values of b for which the x-component of the force on the charge q is a minimum and maximum, we can use Coulomb's Law and vector analysis. The force between two charges is given by:
F = k * (Q * q) / r^2
Where:
- F is the force between the charges
- k is the Coulomb's constant (k = 8.99 x 10^9 N m^2/C^2)
- Q and q are the magnitudes of the charges
- r is the distance between the charges
Let's determine the x-component of the force on q. The x-component can be found by multiplying the force by the cosine of the angle (θ) between the force vector and the x-axis:
Fx = F * cos(θ)
The angle θ can be determined using trigonometry. Considering the configuration of the charges, we can draw a right-angled triangle with the hypotenuse being the distance (r) between the charges, the adjacent side being b, and the opposite side being the y-coordinate of the force (Fy).
Using the Pythagorean theorem, we have:
r^2 = b^2 + (a - Fy)^2
To find the y-coordinate of the force (Fy), we can use the sine function:
sin(θ) = Fy / r
Therefore:
Fy = r * sin(θ) = r * sin(arctan((a-b)/r))
With these equations, we can now find the x-component of the force on q for given values of b.
A) To find the value(s) of b for which the x-component of the force on q is a minimum, we need to differentiate Fx with respect to b and find the critical point(s) where the derivative is equal to zero.
B) Similarly, to find the value(s) of b for which the x-component of the force on q is a maximum, we need to differentiate Fx with respect to b and find the critical point(s) where the derivative does not exist or is not defined, or it exists but is not equal to zero.
By solving the resulting equation(s) or by analyzing the behavior of the differential function, we can find the values of b that correspond to the x-component of the force on q being a minimum or maximum, respectively.