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It is noteworthy to mention an excerpt from Purcell:

The electromotive force was earlier defined as the work per unit charge involved in moving a charge around a circuit containing a voltaic cell. We now broaden the definition of emf to include any influence that causes charge to circulate around a closed path.

More specifically, Feynman writes:

[...] More specifically, emf is the tangential component of the force per unit charge , integrated along the wire once around the circuit. This quantity is equal, therefore, to the total work done on a single charge that travels one around the circuit.

Using Lorentz Force, we get \begin{align}\mathscr E_\textrm{total} \equiv \textrm{Electromotive Force} &=\int_\Gamma (\mathbf E + \mathbf v\times \mathbf B)\,\mathrm d\mathbf s \\&=- \int_{S(t_0)}\, \left(\dfrac{\partial \mathbf B}{\partial t}\right)\bigg|_{t=t_0}\,\mathrm d\mathbf a +\int_{\Gamma}\,(\mathbf v\times \mathbf B)\cdot \mathrm d\mathbf s \\& = \left(-\int_{S(t_0)}\, \left(\dfrac{\partial \mathbf B}{\partial t}\right)\bigg|_{t=t_0}\,\mathrm d\mathbf a\right)+\left(-\dfrac{\mathrm d}{\mathrm dt}\, \int_{\mathrm dS}\mathbf B(t_0)\cdot \mathrm d\mathbf a\right) \\ &=-\dfrac{\mathrm d\Phi_\textrm{total}}{\mathrm dt}\;.\end{align}

It is noteworthy to mention an excerpt from Purcell:

The electromotive force was earlier defined as the work per unit charge involved in moving a charge around a circuit containing a voltaic cell. We now broaden the definition of emf to include any influence that causes charge to circulate around a closed path.

More specifically, Feynman writes:

[...] More specifically, emf is the tangential component of the force per unit charge , integrated along the wire once around the circuit. This quantity is equal, therefore, to the total work done on a single charge that travels one around the circuit.

So, emf is basically the work done by any means that circulate the electron around a closed curve.

In electrostatics, the curl of the electric field was zero since they were conservative and hence there is no non-zero circulation; but in general, the fields are not conservative.

Using Lorentz Force, we get \begin{align}\mathscr E_\textrm{induced} \equiv \textrm{Electromotive Force} &=\int_\Gamma (\mathbf E + \mathbf v\times \mathbf B)\,\mathrm d\mathbf s \\&=- \int_{S(t_0)}\, \left(\dfrac{\partial \mathbf B}{\partial t}\right)\bigg|_{t~=~t_0}\,\mathrm d\mathbf a +\int_{\Gamma}\,(\mathbf v\times \mathbf B)\cdot \mathrm d\mathbf s \\& = \left(-\int_{S(t_0)}\, \left(\dfrac{\partial \mathbf B}{\partial t}\right)\bigg|_{t~=~t_0}\,\mathrm d\mathbf a\right)+\left(-\dfrac{\mathrm d}{\mathrm dt}\, \int_{\mathrm dS}\mathbf B(t_0)\cdot \mathrm d\mathbf a\right) \\ &=-\dfrac{\mathrm d\Phi_\textrm{total}}{\mathrm dt}\;.\end{align}

It is noteworthy to mention an excerpt from Purcell:

The electromotive force was earlier defined as the work per unit charge involved in moving a charge around a circuit containing a voltaic cell. We now broaden the definition of emf to include any influence that causes charge to circulate around a closed path.

More specifically, Feynman writes:

[...] More specifically, emf is the tangential component of the force per unit charge , integrated along the wire once around the circuit. This quantity is equal, therefore, to the total work done on a single charge that travels one around the circuit.

Using Lorentz Force, we get \begin{align}\mathscr E_\textrm{total} \equiv \textrm{Electromotive Force} &=\int_\Gamma (\mathbf E + \mathbf v\times \mathbf B)\,\mathrm d\mathbf s \\&=- \int_{S(t_0)}\, \left(\dfrac{\partial \mathbf B}{\partial t}\right)\bigg|_{t=t_0}\,\mathrm d\mathbf a +\int_{\Gamma}\,(\mathbf v\times \mathbf B)\cdot \mathrm d\mathbf s \\& = \left(-\int_{S(t_0)}\, \left(\dfrac{\partial \mathbf B}{\partial t}\right)\bigg|_{t=t_0}\,\mathrm d\mathbf a\right)+\left(-\dfrac{\mathrm d}{\mathrm dt}\, \int_{\mathrm dS}\mathbf B(t_0)\cdot \mathrm d\mathbf a\right) \;.\end{align}

It is noteworthy to mention an excerpt from Purcell:

The electromotive force was earlier defined as the work per unit charge involved in moving a charge around a circuit containing a voltaic cell. We now broaden the definition of emf to include any influence that causes charge to circulate around a closed path.

More specifically, Feynman writes:

[...] More specifically, emf is the tangential component of the force per unit charge , integrated along the wire once around the circuit. This quantity is equal, therefore, to the total work done on a single charge that travels one around the circuit.

Using Lorentz Force, we get \begin{align}\mathscr E_\textrm{total} \equiv \textrm{Electromotive Force} &=\int_\Gamma (\mathbf E + \mathbf v\times \mathbf B)\,\mathrm d\mathbf s \\&=- \int_{S(t_0)}\, \left(\dfrac{\partial \mathbf B}{\partial t}\right)\bigg|_{t=t_0}\,\mathrm d\mathbf a +\int_{\Gamma}\,(\mathbf v\times \mathbf B)\cdot \mathrm d\mathbf s \\& = \left(-\int_{S(t_0)}\, \left(\dfrac{\partial \mathbf B}{\partial t}\right)\bigg|_{t=t_0}\,\mathrm d\mathbf a\right)+\left(-\dfrac{\mathrm d}{\mathrm dt}\, \int_{\mathrm dS}\mathbf B(t_0)\cdot \mathrm d\mathbf a\right) \\ &=-\dfrac{\mathrm d\Phi_\textrm{total}}{\mathrm dt}\;.\end{align}

It is noteworthy to mention an excerpt from Purcell:

The electromotive force was earlier defined as the work per unit charge involved in moving a charge around a circuit containing a voltaic cell. We now broaden the definition of emf to include any influence that causes charge to circulate around a closed path.

More specifically, Feynman writes:

[...] More specifically, emf is the tangential component of the force per unit charge , integrated along the wire once around the circuit. This quantity is equal, therefore, to the total work done on a single charge that travels one around the circuit.

Using Lorentz Force, we get \begin{align}\mathscr E_\textrm{total} \equiv \textrm{Electromotive Force} &=\int_\Gamma (\mathbf E + \mathbf v\times \mathbf B)\,\mathrm d\mathbf s \\&=- \int_{S(t_0)}\, \left(\dfrac{\partial \mathbf B}{\partial t}\right)\bigg|_{t=t_0}\,\mathrm d\mathbf a +\int_{\Gamma}\,(\mathbf v\times \mathbf B)\cdot \mathrm d\mathbf s \\& = \left(-\int_{S(t_0)}\, \left(\dfrac{\partial \mathbf B}{\partial t}\right)\bigg|_{t=t_0}\,\mathrm d\mathbf a\right)+\left(-\dfrac{\mathrm d}{\mathrm dt}\, \int_{\mathrm dS}\mathbf B(t_0)\cdot \mathrm d\mathbf a\right) \\ &=-\dfrac{\mathrm d\Phi_\textrm{total}}{\mathrm dt}\;.\end{align}

It is noteworthy to mention an excerpt from Purcell:

The electromotive force was earlier defined as the work per unit charge involved in moving a charge around a circuit containing a voltaic cell. We now broaden the definition of emf to include any influence that causes charge to circulate around a closed path.

More specifically, Feynman writes:

[...] More specifically, emf is the tangential component of the force per unit charge , integrated along the wire once around the circuit. This quantity is equal, therefore, to the total work done on a single charge that travels one around the circuit.

Using Lorentz Force, we get \begin{align}\mathscr E_\textrm{total} \equiv \textrm{Electromotive Force} &=\int_\Gamma (\mathbf E + \mathbf v\times \mathbf B)\,\mathrm d\mathbf s \\&=- \int_{S(t_0)}\, \left(\dfrac{\partial \mathbf B}{\partial t}\right)\bigg|_{t=t_0}\,\mathrm d\mathbf a +\int_{\Gamma}\,(\mathbf v\times \mathbf B)\cdot \mathrm d\mathbf s \\& = \left(-\int_{S(t_0)}\, \left(\dfrac{\partial \mathbf B}{\partial t}\right)\bigg|_{t=t_0}\,\mathrm d\mathbf a\right)+\left(-\dfrac{\mathrm d}{\mathrm dt}\, \int_{\mathrm dS}\mathbf B(t_0)\cdot \mathrm d\mathbf a\right) \;.\end{align}