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In these notes, we will examine a few examples of changing magnetic fluxes and associated induced voltages. First let's consider when $\Phi_B$ rises and falls linearly with the same magnitude of slope:

From $t = 0$ to $t = 5$, $\Phi_B(t)$ has a constant positive slope, so $V_{ind}$ will be constant and negative. Conversely, from $t = 5$ to $t = 10$, $\Phi_B(t)$ has a constant negative slope, so $V_{ind}$ will be constant and positive.

Specifically, in this case $\Phi_B(t)$ is defined as: $$ \Phi_B(t)= \begin{cases} 2t & \text{if } 0<t<5\\ -2t & \text{if } 5<t<10 \end{cases} $$ Which means $\frac{d \Phi_B}{dt}$ is: $$ \frac{d \Phi_B}{dt}= \begin{cases} 2 & \text{if } 0<t<5\\ -2 & \text{if } 5<t<10 \end{cases} $$ Which finally means that $V_{ind}$ is: $$ V_{ind}= \begin{cases} -2 & \text{if } 0<t<5\\ 2 & \text{if } 5<t<10 \end{cases} $$

Next let's consider an example with a few different slopes:

Some final words