Identical +1.45 µC charges are fixed to adjacent corners of a square. What charge (magnitude and algebraic sign) should be fixed to one of the empty corners, so that the total electric potential at the remaining empty corner is 0 V?
Well, in order to find the charge needed at the remaining empty corner, let's break out our trusty clown calculator!
*Clown Bot pulls out a ridiculous-looking calculator with a red nose.*
Now let's see... we have identical positive charges at two corners, so we need a negative charge at the remaining corner to balance things out.
*Clown Bot hits some buttons on the calculator, which makes funny loud noises.*
Calculating... calculating... Ah-ha! According to my clown calculations, the magnitude of the charge needed should be 1.45 µC, but it should be negative, since it needs to balance out the positive charges.
So, in clown terms, the charge you need at the empty corner is -1.45 µC. That should make the total electric potential at the remaining empty corner equal to zero volts and keep everything in clown-tastic balance!
*Clown Bot does a little dance while juggling imaginary charges.*
To find the charge that should be fixed to one of the empty corners so that the total electric potential at the remaining empty corner is 0 V, we can use the principle of superposition and the equation for electric potential.
Let's label the corners of the square as A, B, C, and D. Charges +1.45 µC are fixed at corners A and B, and we need to find the charge at corner C.
1. Calculate the electric potential at corner C due to the charge at corner A.
- The electric potential V1 at C due to the charge at A can be calculated using the equation:
V1 = k*qA / r1
where k is the Coulomb's constant (9 x 10^9 Nm^2/C^2), qA is the charge at A (+1.45 µC), and r1 is the distance between A and C (the length of one side of the square).
2. Calculate the electric potential at corner C due to the charge at corner B.
- The electric potential V2 at C due to the charge at B can also be calculated using the same equation as above:
V2 = k*qB / r2
where qB is the charge at B (+1.45 µC), and r2 is the distance between B and C (the length of one side of the square).
3. The total electric potential at C, Vtotal, is the sum of the potentials due to the charges at A and B:
Vtotal = V1 + V2
4. Set Vtotal equal to 0 V and solve for the charge at C, qC.
0 = V1 + V2
qC = - (qA*r1 + qB*r2) / r3
where r3 is the distance between A and C (or B and C).
Therefore, the magnitude and algebraic sign of the charge at C should be -(qA*r1 + qB*r2) / r3.
Note: Make sure to use consistent units for all the values in the equations (e.g., convert µC to C if needed, and meters for distances).
To find the charge that should be fixed to one of the empty corners, we can use the concept of electric potential.
Given that the charges on the two adjacent corners of the square are +1.45 µC each, let's call this charge Q1.
To achieve a total electric potential of 0 V at one of the remaining empty corners, we need to find the charge, let's call it Q2, that should be fixed there.
The electric potential at a point due to a charge is given by the equation:
V = k * Q / r
Where:
V is the electric potential at the point,
k is a constant called the electrostatic constant or Coulomb's constant (k ≈ 8.99 * 10^9 Nm²/C²),
Q is the charge creating the electric potential, and
r is the distance from the charge.
Since we have two charges, Q1 and Q2, creating the electric potential, the total electric potential at the remaining empty corner is the sum of the electric potentials due to each charge:
V_total = V1 + V2
Given that V_total = 0 V and V1 = k * Q1 / r1, we can rearrange the equation to solve for Q2:
Q2 = -V1 * r2 / (k * r1)
Where:
Q2 is the charge that needs to be fixed to one of the empty corners,
V1 is the electric potential due to Q1,
r2 is the distance from the empty corner to Q2, and
r1 is the distance between the charged corners.
To calculate the values, we need to know the distances between the corners. Let's assume that the distance between the charged corners is denoted by 'd'.
Therefore, the charge Q2 that should be fixed to one of the empty corner is given by:
Q2 = - (k * Q1 * r2) / (k * r1 * d)
In this case, Q1 = 1.45 µC and r1 = d.
Thus, the charge (magnitude and algebraic sign) that should be fixed to one of the empty corners is determined by substituting the given values into the equation.