1) A point charge q2 = -0.5 μC is fixed at the origin of a co-ordinate system as shown. Another point charge q1 = 0.6 μC is initially located at point P, a distance d1 = 8.9 cm from the origin along the x-axis.
a. What is ΔPE, the change in potenial energy of charge q1 when it is moved from point P to point R, located a distance d2 = 3.6 cm from the origin along the x-axis as shown?
b. The charge q2 is now replaced by two charges q3 and q4 which each have a magnitude of -0.25 μC, half of that of q2. The charges are located a distance a = 2.2 cm from the origin along the y-axis as shown. What is ΔPE, the change in potential energy now if charge q1 is moved from point P to point R?
c. What is the potential energy of the system composed of the three charges q1, q3, and q4, when q1 is at point R? Define the potential energy to be zero at infinity.
d. The charge q4 is now replaced by charge q5 which has the same magnitude, but opposite sign from q4 (i.e., q5 = 0.25 μC). What is the new value for the potential energy of the system?
a. ΔPE = -2.7 x 10^-6 J
b. ΔPE = -1.3 x 10^-6 J
c. PE = -3.9 x 10^-6 J
d. PE = -2.6 x 10^-6 J
a. To calculate the change in potential energy of charge q1 when it is moved from point P to point R, we need to consider the work done by the electric field. The formula for the change in potential energy is given by:
ΔPE = -W
where W is the work done by the electric field. The work done can be calculated using the formula:
W = q1 * ΔV
where q1 is the charge and ΔV is the change in electric potential.
First, we need to calculate the electric potential at point P and at point R. The electric potential at a point due to a point charge is given by the formula:
V = k * q2 / r
where k is the Coulomb's constant (k = 8.99 x 10^9 Nm^2/C^2), q2 is the charge, and r is the distance from the origin.
At point P, the distance from the origin is d1 = 8.9 cm = 0.089 m, and at point R, the distance from the origin is d2 = 3.6 cm = 0.036 m.
So, the electric potential at point P is:
Vp = k * q2 / d1
And the electric potential at point R is:
Vr = k * q2 / d2
Next, we calculate the change in electric potential:
ΔV = Vr - Vp
Finally, we calculate the change in potential energy:
ΔPE = -q1 * ΔV
Substituting the values of q1, ΔV, and the charge values into the equation will give the answer.
b. In this scenario, we have two charges q3 and q4, each with a magnitude of -0.25 μC. The charges are located at a distance a = 2.2 cm = 0.022 m from the origin along the y-axis.
To calculate the change in potential energy when q1 is moved from point P to point R, we first need to calculate the electric potential due to charges q3 and q4 at point P and at point R.
The electric potential due to a point charge at a distance r from the origin is given by the formula:
V = k * q / r
where k is the Coulomb's constant, q is the charge, and r is the distance from the origin.
At point P, the distance from charges q3 and q4 is given by:
d3 = √(d1^2 + a^2)
And at point R, the distance from charges q3 and q4 is given by:
d4 = √(d2^2 + a^2)
So, the electric potential at point P due to charges q3 and q4 is:
Vp = k * (q3 + q4) / d3
And the electric potential at point R due to charges q3 and q4 is:
Vr = k * (q3 + q4) / d4
Next, we calculate the change in electric potential:
ΔV = Vr - Vp
Finally, we calculate the change in potential energy:
ΔPE = -q1 * ΔV
Substituting the values of q1, ΔV, and the charge values into the equation will give the answer.
c. To calculate the potential energy of the system composed of charges q1, q3, and q4 when q1 is at point R, we need to consider the potential energy due to each pair of charges.
The potential energy between two point charges is given by the formula:
PE = k * (q1 * q2 / r)
where k is the Coulomb's constant, q1 and q2 are the charges, and r is the distance between the charges.
So, the potential energy between charges q1 and q3 is:
PE13 = k * (q1 * q3 / d2)
And the potential energy between charges q1 and q4 is:
PE14 = k * (q1 * q4 / d2)
Finally, we sum up these potential energies to get the total potential energy:
PE_total = PE13 + PE14
Substituting the values of q1, q3, q4, and d2 into the equation will give the answer.
d. In this scenario, charge q4 is replaced by charge q5, which has the same magnitude but opposite sign (q5 = -0.25 μC).
To calculate the new value for the potential energy of the system, we need to adjust the potential energy calculation for the new charge.
The potential energy between two point charges with opposite signs is given by the formula:
PE = -k * (q1 * q2 / r)
So, the potential energy between charges q1 and q5 is:
PE15 = -k * (q1 * q5 / d2)
Finally, we sum up the potential energies to get the total potential energy:
PE_total = PE13 + PE15
Substituting the values of q1, q3, q5, and d2 into the equation will give the new value for the potential energy of the system.
a. To find the ΔPE, the change in potential energy of charge q1 when it is moved from point P to point R, we can use the formula for potential energy:
ΔPE = PE_final - PE_initial
To find the potential energy at point P, we need to calculate the electric potential due to q2 at that point. The formula for electric potential is given by:
PE = k * (q1 * q2) / r
Where k is the Coulomb's constant (8.99 * 10^9 N m^2/C^2), q1 is the charge at point P (0.6 μC), q2 is the charge at the origin (-0.5 μC), and r is the distance between the charges.
Using the given values, we can calculate the potential energy at point P:
PE_initial = k * (q1 * q2) / d1
Next, we need to calculate the potential energy at point R. To do this, we need to use the same formula but with the new distance d2:
PE_final = k * (q1 * q2) / d2
Finally, we can calculate the change in potential energy:
ΔPE = PE_final - PE_initial
Plug in the values and calculate the answer.
b. To find the change in potential energy now if charge q1 is moved from point P to point R, we need to consider the new charges q3 and q4. Since they have the same magnitude as q2 but with opposite signs, their electric potentials add up.
The formula for potential energy now becomes:
PE = k * [(q1 * q2) / r1 + (q1 * q3) / r2 + (q1 * q4) / r3]
Where r1 is the distance between charge q2 and q1, r2 is the distance between charge q3 and q1, and r3 is the distance between charge q4 and q1.
Using the given values, we can calculate the potential energy at point P and point R using the same formula:
PE_initial = k * [(q1 * q2) / d1 + (q1 * q3) / sqrt(a^2 + d1^2)]
PE_final = k * [(q1 * q2) / d2 + (q1 * q3) / sqrt(a^2 + d2^2)]
Then, we can calculate the change in potential energy:
ΔPE = PE_final - PE_initial
Plug in the values and calculate the answer.
c. To find the potential energy of the system composed of the three charges q1, q3, and q4 when q1 is at point R, we need to consider the electric potentials due to each charge:
PE = k * [(q1 * q2) / d2 + (q1 * q3) / sqrt(a^2 + d2^2) + (q1 * q4) / sqrt(a^2 + d2^2)]
Plug in the given values and calculate the answer.
d. To find the new value for the potential energy of the system when charge q4 is replaced by charge q5, we need to consider the changes in the electric potentials:
PE_new = k * [(q1 * q2) / d2 + (q1 * q3) / sqrt(a^2 + d2^2) + (q1 * q5) / sqrt(a^2 + d2^2)]
Plug in the given values and calculate the answer.