# Calculating Energy Dissipated in Copper Wire Loop Due to Magnetic Field Increase

• StudentofPhysics
In summary, the formula for calculating energy dissipated in a copper wire loop due to magnetic field increase is: E = 0.5 * L * I^2 * B^2 * R. The units for the variables in the formula are Joules (J) for energy, meters (m) for length, Amperes (A) for current, Tesla (T) for magnetic field, and Ohms (Ω) for resistance. The length of the wire directly affects the amount of energy dissipated, with longer wires resulting in higher energy dissipation due to the higher resistance. The relationship between current and energy dissipation is squared, meaning that higher currents result in exponentially higher energy dissipation. Similarly, the increase in
StudentofPhysics
A piece of copper wire is formed into a single circular loop of radius 9 cm. A magnetic field is oriented parallel to the normal to the loop, and it increases from 0 to 0.45 T in a time of 0.45 s. The wire has a resistance per unit length of 3.3 10-2 /m. What is the average electrical energy dissipated in the resistance of the wire.

I know the emf = 0.254 Volts.

How do I find the energy dissipated?

Start by finding the current that flows through the wire.

As a scientist, it is important to understand the relationship between magnetic fields and electrical currents. In this scenario, the increasing magnetic field induces a current in the copper wire loop, which in turn causes the wire to dissipate energy in the form of heat due to its resistance. To calculate the average electrical energy dissipated in the resistance of the wire, we can use the formula:

E = I^2Rt

Where E is the energy dissipated, I is the current, R is the resistance per unit length, and t is the time.

First, we need to calculate the current (I) in the wire using Ohm's law:

I = V/R

Where V is the emf (0.254 V) and R is the resistance per unit length (3.3 x 10^-2 Ω/m).

Substituting these values, we get I = 0.254 V / 3.3 x 10^-2 Ω/m = 7.7 A.

Next, we need to calculate the total resistance of the wire loop. Since it is a circular loop, we can use the formula for the resistance of a wire in a circular loop:

R = ρL/A

Where ρ is the resistivity of copper (1.68 x 10^-8 Ωm), L is the length of the wire (circumference of the loop = 2πr = 2π(0.09 m) = 0.565 m), and A is the cross-sectional area of the wire (πr^2 = π(0.09 m)^2 = 0.0257 m^2).

Substituting these values, we get R = (1.68 x 10^-8 Ωm)(0.565 m) / 0.0257 m^2 = 3.7 x 10^-6 Ω.

Now, we can plug in these values into the original formula to calculate the energy dissipated:

E = (7.7 A)^2 (3.7 x 10^-6 Ω) (0.45 s) = 1.5 x 10^-4 J

Therefore, the average electrical energy dissipated in the resistance of the wire is 1.5 x 10^-4 Joules. This energy is converted into heat, which can be measured by the temperature increase of the wire.

## What is the formula for calculating energy dissipated in a copper wire loop due to magnetic field increase?

The formula for calculating energy dissipated in a copper wire loop due to magnetic field increase is:
E = 0.5 * L * I^2 * B^2 * R, where E represents energy dissipated, L is the length of the wire, I is the current flowing through the wire, B is the increase in magnetic field, and R is the resistance of the wire.

## What are the units for the variables in the energy dissipation formula?

The units for the variables in the energy dissipation formula are:
E: Joules (J)
L: meters (m)
I: Amperes (A)
B: Tesla (T)
R: Ohms (Ω)

## How does the length of the wire affect the amount of energy dissipated?

The length of the wire directly affects the amount of energy dissipated. As the length of the wire increases, the amount of energy dissipated also increases. This is because a longer wire has a higher resistance, which results in more energy being converted into heat.

## What is the relationship between the current and the amount of energy dissipated?

The current flowing through the wire has a squared relationship with the amount of energy dissipated. This means that as the current increases, the amount of energy dissipated increases exponentially. Therefore, it is important to use caution when handling high currents to prevent excessive energy dissipation and potential hazards.

## How does the change in magnetic field affect the amount of energy dissipated?

The increase in magnetic field has a squared relationship with the amount of energy dissipated. This means that as the magnetic field increases, the amount of energy dissipated also increases exponentially. Therefore, it is important to carefully monitor and control the magnetic field to prevent excessive energy dissipation and potential hazards.

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