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The electromotive force (emf) is a fundamental concept in understanding how electrical circuits operate. It is defined as the voltage developed by any source of electrical energy, like a battery or generator. When it comes to a coil, emf is the voltage generated when there's a change in the magnetic field within the coil.
In this exercise, the emf induced in a coil is calculated using the formula: \[ \varepsilon = -L \frac{\Delta I}{\Delta t} \]where:

• \( \varepsilon \) is the induced emf,
• \( L \) is the inductance,
• \( \Delta I \) is the change in current,
• \( \Delta t \) is the change in time.

This formula is derived from Faraday's law of electromagnetic induction, which tells us how a change in magnetic environment of a coil will induce an electromotive force.
Remember, the negative sign in the formula indicates that the induced emf always opposes the change in current, a principle known as Lenz's law.

Understanding the rate of change of current is crucial for calculating electromagnetic effects accurately. In this problem, the rate of change of current (\( \frac{\Delta I}{\Delta t} \)) represents how fast the current changes over a particular period.
To find it:

• Current change \( \Delta I \) is found by subtracting the initial current from the final current, converted from milliamps to amps for accuracy.

So, the change was calculated as:\[ \Delta I = (+0.025 \mathrm{\,A}) \,-\, (-0.028 \mathrm{\,A}) = 0.053 \mathrm{\,A} \]This tells us there is a significant change in current across the coil.
Next, we calculate the time interval in seconds from milliseconds:

• \( \Delta t = 12.0 \mathrm{\,ms} = 0.012 \mathrm{\,s} \)

Thus, the rate of change of current is:\[ \frac{0.053}{0.012} \approx 4.42 \mathrm{\,A/s} \]This value is vital in determining the impact of current change on the coil's inductance.

Converting units is an essential skill for working through physics and engineering problems. In this exercise, several conversions were necessary to align all values with the standard International System of Units (SI units) used in scientific computations.
The currents given were initially in milliamps (mA). Since the standard unit for current is amperes (A), conversions were required:

• Convert \( +25.0 \mathrm{\,mA} \) to amperes: \( +25.0 \mathrm{\,mA} = +0.025 \mathrm{\,A} \).
• Convert \( -28.0 \mathrm{\,mA} \) to amperes: \( -28.0 \mathrm{\,mA} = -0.028 \mathrm{\,A} \).

This ensures consistency across calculations.
Similarly, the time interval was given in milliseconds (ms). It needed to be converted into seconds (s), since seconds is the SI unit for time:

• \( 12.0 \mathrm{\,ms} = 0.012 \mathrm{\,s} \).

These conversions facilitate using formulas aptly, ensuring precise results that comply with scientific standards.