Two point charges with q1 = +3 .2×^10−7 C and q2 = −4.8×10^−7 C are initially separated by a distance of r =2 .4×10^−4 m.
(a) How much energy is required to double their separation?
(b) What is the electric potential at the point midway between the two charges when their separation is 2r?
b) The potential is a scalar, and can simply be added up.
Total Potential = k(q1)/(r1) + k(q2)/(r2)
At the point midway, both distances are half of 2r, that is, r1 = r2 = r
=> Potential = k((q1)/(r) + (q2)/r))
= (k/r)((3.2 - 4.8)*10^-7)
= -(9*10^9)(1.6*10^-7)/(2.4*10^-4)
= -6*10^6 V
a)
Initial potential energy of the system =
k(q1)(q2)/r
Final potential energy of the system =
k(q1)(q2)/2r
Work done = Energy Required = Loss in potential Energy of the system
k(q1)(q2)/r - k(q1)(q2)/2r
= k(q1)(q2)/2r
= (9*10^9)*(3.2*10^-7)*(4.8*10^-7)/4.8*10^-4
= 2.88J
https://www.youtube.com/watch?v=VmpihFsziFw
in a) Why are you dividing by 2r when that change in seperation is r?
(a) How much energy is required to double their separation?
Well, you know what they say, distance makes the heart grow fonder! But in this case, we're talking about charges, not hearts. To find the energy required to double their separation, we can use the formula for electric potential energy:
U = k * q1 * q2 / r
Where U is the potential energy, k is the constant of proportionality (approximately equal to 9 x 10^9 N m^2/C^2), q1 and q2 are the charges, and r is the separation distance.
So, to find the energy required to double their separation, we'll need to double the value of r:
U_new = k * q1 * q2 / (2 * r)
Just plug in the values and do some calculations, and you'll find your answer!
(b) What is the electric potential at the point midway between the two charges when their separation is 2r?
Ah, the point midway between two charges - it's like being caught between a rock and a hard place! But fear not, calculating the electric potential at that point is not too hard.
The electric potential at any point due to two point charges can be calculated using the formula:
V = k * (q1 / r1 + q2 / r2)
Where V is the electric potential, k is still the constant of proportionality, q1 and q2 are the charges, and r1 and r2 are the distances from the charges to the point.
In this case, since the charges are symmetrically placed, the distances r1 and r2 will be equal. So, when the separation is 2r, you can substitute r1 = r2 = r into the formula to find the electric potential.
Just plug in the values, do some calculations, and ta-da! You'll have your answer! Don't worry, you won't be caught between a rock and a hard place for too long!
To find the answers to these questions, we can use the formulas for electric potential energy and electric potential.
(a) To calculate the energy required to double the separation, we first need to find the new separation. Doubling the separation means multiplying the original separation by 2. So the new separation, r', is given by:
r' = 2r = 2 * 2.4×10^−4 m = 4.8×10^−4 m
Now, we can calculate the potential energy using the formula:
U = k * (q1 * q2) / r'
where k is the electrostatic constant, q1 and q2 are the magnitudes of the two charges, and r' is the new separation.
Plugging in the values:
k = 8.99 × 10^9 N m^2/C^2 (electrostatic constant)
q1 = +3.2 × 10^−7 C (charge of q1)
q2 = −4.8 × 10^−7 C (charge of q2)
r' = 4.8 × 10^−4 m (new separation)
U = (8.99 × 10^9 N m^2/C^2) * ((+3.2 × 10^−7 C) * (−4.8 × 10^−7 C)) / (4.8 × 10^−4 m)
Calculating this expression will give you the energy required to double their separation.
(b) To find the electric potential at the point midway between the two charges when their separation is 2r, we can use the formula:
V = k * (q1 / r1) + k * (q2 / r2)
where V is the electric potential, k is the electrostatic constant, q1 and q2 are the magnitudes of the two charges, and r1 and r2 are the distances from the midpoint to each charge.
In this case, since the separation is 2r, the distances r1 and r2 are equal to r. Hence, the formula becomes:
V = k * (q1 / r) + k * (q2 / r)
Plugging in the values:
k = 8.99 × 10^9 N m^2/C^2 (electrostatic constant)
q1 = +3.2 × 10^−7 C (charge of q1)
q2 = −4.8 × 10^−7 C (charge of q2)
r = 2.4 × 10^−4 m (original separation)
V = (8.99 × 10^9 N m^2/C^2) * ((+3.2 × 10^−7 C) / (2.4 × 10^−4 m)) + (8.99 × 10^9 N m^2/C^2) * ((−4.8 × 10^−7 C) / (2.4 × 10^−4 m))
Calculating this expression will give you the electric potential at the midpoint when their separation is 2r.