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Engineering LibreTexts

5.11: Kirchoff’s Voltage Law for Electrostatics: Differential Form

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  • The integral form of Kirchoff’s Voltage Law for electrostatics (KVL; Section [m0016_Kirchoffs_Voltage_Law_Electrostatics_Integral_Form]) states that an integral of the electric field along a closed path is equal to zero: \[\oint_{\mathcal C}{ {\bf E} \cdot d{\bf l} } = 0\] where \({\bf E}\) is electric field intensity and \({\mathcal C}\) is the closed curve. In this section, we derive the differential form of this equation. In some applications, this differential equation, combined with boundary conditions imposed by structure and materials (Sections [m0020_Boundary_Conditions_on_E] and [m0021_Boundary_Conditions_on_D]), can be used to solve for the electric field in arbitrarily complicated scenarios. A more immediate reason for considering this differential equation is that we gain a little more insight into the behavior of the electric field, disclosed at the end of this section.

    The equation we seek may be obtained using Stokes’ Theorem (Section [m0051_Stokes_Theorem]), which in the present case may be written: \[\int_{\mathcal S} \left( \nabla \times {\bf E} \right) \cdot d{\bf s} = \oint_{\mathcal C} {\bf E}\cdot d{\bf l} \label{m0152_eKVL2}\] where \({\mathcal S}\) is any surface bounded by \({\mathcal C}\), and \(d{\bf s}\) is the normal to that surface with direction determined by right-hand rule. The integral form of KVL tells us that the right hand side of the above equation is zero, so: \[\int_{\mathcal S} \left( \nabla \times {\bf E} \right) \cdot d{\bf s} = 0\] The above relationship must hold regardless of the specific location or shape of \({\mathcal S}\). The only way this is possible for all possible surfaces is if the integrand is zero at every point in space. Thus, we obtain the desired expression: \[\boxed{ \nabla \times {\bf E} = 0 } \label{m0152_eKVL}\] Summarizing:

    The differential form of Kirchoff’s Voltage Law for electrostatics (Equation [m0152_eKVL]) states that the curl of the electrostatic field is zero.

    Equation [m0152_eKVL] is a partial differential equation. As noted above, this equation, combined with the appropriate boundary conditions, can be solved for the electric field in arbitrarily-complicated scenarios. Interestingly, it is not the only such equation available for this purpose – Gauss’ Law (Section [m0045_Gauss_Law_Differential_Form_from_Integral_Form]) also does this. Thus, we see a system of partial differential equations emerging, and one may correctly infer that that the electric field is not necessarily fully constrained by either equation alone.

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