# A charge of #-1 C# is at the origin. How much energy would be applied to or released from a # 6 C# charge if it is moved from # (-3, 1 ) # to #(4 ,4 ) #?

The energy released is

Potential energy is

The separation

The separation

Consequently,

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The energy applied to or released from a charge when it is moved in an electric field can be calculated using the formula:

ΔU = q * ΔV

Where: ΔU is the change in potential energy, q is the charge being moved, and ΔV is the change in electric potential.

To find ΔV, we can use the formula for the electric potential difference between two points in space:

ΔV = V_final - V_initial

Given that the charge at the origin creates an electric potential, and we're moving the 6 C charge from (-3, 1) to (4, 4), we need to calculate the electric potential at both points using the formula:

V = k * (Q / r)

Where: k is Coulomb's constant (8.99 x 10^9 Nm^2/C^2), Q is the charge creating the electric potential, and r is the distance from the charge to the point where the electric potential is being measured.

First, let's calculate the electric potential at the initial point (-3, 1): r_initial = √((-3 - 0)^2 + (1 - 0)^2) = √(9 + 1) = √10 V_initial = (8.99 x 10^9 Nm^2/C^2) * (-1 C) / √10 ≈ -2.83 x 10^9 V

Next, let's calculate the electric potential at the final point (4, 4): r_final = √((4 - 0)^2 + (4 - 0)^2) = √(16 + 16) = √32 V_final = (8.99 x 10^9 Nm^2/C^2) * (-1 C) / √32 ≈ -1.59 x 10^9 V

Now, let's find the change in electric potential (ΔV): ΔV = V_final - V_initial ΔV = -1.59 x 10^9 V - (-2.83 x 10^9 V) = 1.24 x 10^9 V

Finally, let's find the change in potential energy (ΔU): ΔU = q * ΔV ΔU = 6 C * 1.24 x 10^9 V = 7.44 x 10^9 J

So, approximately 7.44 x 10^9 joules of energy would be applied to or released from the 6 C charge when it is moved from (-3, 1) to (4, 4).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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