The induced voltage versus time for a coil that has 100...

The induced voltage versus time for a coil that has 100 circular turns of wire with radii $25 \mathrm{~cm}$ is given in the table. Plot $V(t)$ and use this graph to predict the graph of the magnetic field as a function of time $B(t)$ that is passing through the loop (assume the magnetic field is perpendicular to the plane of the loop). $$ \begin{array}{lcrc} \boldsymbol{t}(\mathrm{s}) & \boldsymbol{V}(\mathbf{V}) & \boldsymbol{t}(\mathbf{s}) & \boldsymbol{V}(\mathbf{V}) \\ \hline 0 & 0 & 9 & 2 \\ 1 & 2 & 10 & 4 \\ 2 & 4 & 11 & 2 \\ 3 & 2 & 12 & 0 \\ 4 & 0 & 13 & -2 \\ 5 & -2 & 14 & -4 \\ 6 & -4 & 15 & -2 \\ 7 & -2 & 16 & 0 \\ 8 & 0 & & \\ \end{array} $$ $$ \begin{array}{lrll} \hline \end{array} $$

The induced voltage versus time for a coil that has 100 circular turns of wire with radii $25 \mathrm{~cm}$ is given in the table. Plot $V(t)$ and use this graph to predict the graph of the magnetic field as a function of time $B(t)$ that is passing through the loop (assume the magnetic field is perpendicular to the plane of the loop).
$$
\begin{array}{lcrc}
\boldsymbol{t}(\mathrm{s}) & \boldsymbol{V}(\mathbf{V}) & \boldsymbol{t}(\mathbf{s}) & \boldsymbol{V}(\mathbf{V}) \\
\hline 0 & 0 & 9 & 2 \\
1 & 2 & 10 & 4 \\
2 & 4 & 11 & 2 \\
3 & 2 & 12 & 0 \\
4 & 0 & 13 & -2 \\
5 & -2 & 14 & -4 \\
6 & -4 & 15 & -2 \\
7 & -2 & 16 & 0 \\
8 & 0 & & \\
\end{array}
$$
$$
\begin{array}{lrll}

\hline
\end{array}
$$

Key Concepts

Faraday's Law of Electromagnetic Induction

This fundamental principle states that a changing magnetic flux through a circuit induces an electromotive force (EMF) in the circuit. It is mathematically expressed as the induced EMF being equal to the negative rate of change of the magnetic flux, and it underpins the relationship between the voltage measured in the coil and the time variation of the magnetic field passing through it.

Magnetic Flux

Magnetic flux is a measure of the total magnetic field passing through a given area. It is calculated as the product of the magnetic field strength, the area of the loop, and the cosine of the angle between the field and the normal to the plane of the loop. This concept is key because changes in magnetic flux are directly related to the induced voltage in the coil.

Coil Geometry and Area

The area of the coil, which depends on its geometry (in this case, the radius of the circular loops), is essential when relating the magnetic field and the magnetic flux. A larger area results in greater magnetic flux for the same magnetic field strength, thereby having a proportionate effect on the induced voltage measured in the coil.

Effect of Multiple Turns

When a coil consists of multiple turns, the total induced voltage is amplified by the number of turns. This is because the induced EMF in each individual turn adds up, making the total EMF equal to the number of turns multiplied by the EMF induced in one turn. This factor is critical when determining the relationship between the measured EMF and the magnetic field that causes the flux change.

Integration to Determine Magnetic Field

Since the induced voltage is proportional to the time derivative of the magnetic flux (and hence the magnetic field for constant area and orientation), integrating the voltage with respect to time allows one to recover the magnetic flux and thus the magnetic field. This integration essentially reveals the underlying time variation of the magnetic field based on the voltage signal, shaping its graph accordingly.

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