I'm trying to physically understand the responses of this pair of (dual) circuits: a step current source driving a capacitor and a step voltage source driving an inductor.

^{simulate this circuit – Schematic created using CircuitLab}

The capacitor output voltage is a ramp and the inductor output current is a ramp, as expected. I understand fine how to mathematically derive these responses, what I'm trying to do is physically explain them.

What I have for the capacitor is that if we assume zero initial charge on the capacitor, then its initial voltage is zero because there is no charge difference. At time t = 0 the current source steps to some \$I_0\$. This means that, by definition of current, the source is now transferring charge onto the plates of the capacitor such that a charge separation develops on the plates over time. Assuming an LTI capacitor such that \$ q = CV \$, the voltage ramps up since the charge difference on its plates is being increased at a constant rate.

I can't quite articulate an equivalent description of the inductor circuit. I have a basic understanding of the phyiscal origin/operation of an inductor but I'm struggling to put it together here. My understanding is that, assuming an LTI inductor, it is defined by the relationship \$ \phi = Li \$, where \$ \phi \$ is the total flux linked through in the inductor. I also know that by Faraday's law, \$ v(t) = -\frac{d\phi}{dt} \$, so we can differentiate the inductor relation to get \$ v(t) = L \frac{di(t)}{dt} \$ (dropped the minus sign). From here I can mathematically derive current as a function of voltage, but why is this constant (step) current source apparently causing a constant change in magnetic flux that causes the finite change in current?

When I think of a change in magnetic flux through a current loop that induces an EMF and a current, I think of something like this: I don't see how the voltage source creating a constant voltage across the inductor's terminals does this.

I'm trying to physically understand the responses of this pair of (dual) circuits: a step current source driving a capacitor and a step voltage source driving an inductor.

^{simulate this circuit – Schematic created using CircuitLab}

The capacitor output voltage is a ramp and the inductor output current is a ramp, as expected. I understand fine how to mathematically derive these responses, what I'm trying to do is physically explain them.

What I have for the capacitor is that if we assume zero initial charge on the capacitor, then its initial voltage is zero because there is no charge difference. At time t = 0 the current source steps to some \$I_0\$. This means that, by definition of current, the source is now transferring charge onto the plates of the capacitor such that a charge separation develops on the plates over time. Assuming an LTI capacitor such that \$ q = CV \$, the voltage ramps up since the charge difference on its plates is being increased at a constant rate.

I can't quite articulate an equivalent description of the inductor circuit. I have a basic understanding of the phyiscal origin/operation of an inductor but I'm struggling to put it together here. My understanding is that, assuming an LTI inductor, it is defined by the relationship \$ \phi = Li \$, where \$ \phi \$ is the total flux linked through in the inductor. I also know that by Faraday's law, \$ v(t) = -\frac{d\phi}{dt} \$, so we can differentiate the inductor relation to get \$ v(t) = L \frac{di(t)}{dt} \$ (dropped the minus sign). From here I can mathematically derive current as a function of voltage, but why is this constant (step) voltage source apparently causing a constant change in magnetic flux that causes the finite change in current?

When I think of a change in magnetic flux through a current loop that induces an EMF and a current, I think of something like this: I don't see how the voltage source creating a constant voltage across the inductor's terminals does this.