Electrical properties of an L-shaped conducting plate

Ivchenko, Vladimir V.

doi:10.1590/1806-9126-RBEF-2025-0510

Abstract

Conducting elements with sharp bends and non-uniform geometry are common in printed circuit boards, thin-film devices, and microelectronic interconnects, where accurate evaluation of current distribution and DC resistance is essential. An L-shaped conducting plate represents a simple yet non-trivial example of such geometries, for which standard analytical formulas are not applicable. In this work, a two-dimensional boundary value problem for the Laplace equation with mixed Dirichlet–Neumann boundary conditions is solved numerically using the finite element method. The resulting electric potential is used to compute the electric field distribution and the total current, from which the DC resistance of the plate is obtained. The numerical results reveal strong current crowding and a singular enhancement of the electric field near the inner right-angle corner, accompanied by a pronounced field reduction at the outer corner. Due to this non-uniform current distribution, the total DC resistance of the L-shaped plate is found to be systematically lower than that of the corresponding straightened strip and decreases monotonically with increasing transverse-to-length aspect ratio. The problem thus provides a clear and physically transparent example of how non-trivial geometry affects electrical resistance, making it valuable both for practical applications and for advanced undergraduate education.

Keywords:
L-shaped plate; DC resistance; finite element method; field singularity; mesh convergence

1. Introduction

An L-shaped conductive plate is a geometry frequently encountered in engineering applications where current distribution, heat dissipation, and electrical resistance are critical factors. Accurate calculation of its direct current (DC) resistance is essential for modelling current flow in irregular geometries, where standard analytical formulas for straight conductors are not applicable [¹]. This typically requires numerical techniques such as the finite element method, conformal mapping, or Monte Carlo simulations.

L-shaped conductive paths are widely used for signal and power routing in compact electronic circuits. Knowledge of DC resistance enables estimation of voltage drops, thermal dissipation, and overall reliability. In high-current applications, this analysis helps to prevent local overheating and electro-migration in corner regions with increased current density [²].

In microelectronics, L-shaped resistors made of poly-silicon or metal films are employed to optimize chip area utilization. Precise resistance calculation is crucial for parameter tuning in analog circuits and sensors, as well as for process control during semiconductor fabrication, where sheet resistance serves as a key metric for thin-film quality. Similar considerations apply to embedded resistors in LTCC (low-temperature co-fired ceramic) substrates used in sensor and amplifier designs.

The L-shaped plate also serves as a fundamental model for studying how geometrical parameters, such as angle and segment length, influence contact resistance. In applied electromagnetics, L-shaped grounding electrodes are used to improve current dissipation into the earth [³]. Moreover, L-shaped electrode configurations appear in electrostatic systems – for example, in compact deflection arrays of synchronous ink-jet printers, where overlapping L-shaped boundaries define the electric field distribution [⁴].

From a theoretical standpoint, L-shaped domains provide a benchmark problem for testing numerical and analytical methods in potential theory. The Laplacian problem over such a domain can be solved using singular-function boundary integral formulations [⁵], while charge distribution can be analyzed via last-passage Monte Carlo algorithms. Electrokinetic transport through L-shaped micro-channels under DC fields has also been extensively studied [⁶], both experimentally and theoretically, illustrating the broader relevance of this geometry across physical and engineering contexts.

Our experience shows that many students try to calculate the resistance of an L-shaped plate by replacing it with a rectified analogue and applying known formulas for a straight conductor with current. In fact, the electric field in the plate is non-uniform and changes from point to point both in absolute magnitude and in direction. At the same time, some other students repeatedly asked how the deviations from field homogeneity should modify the calculation of the electrical resistance. Some aspects of this issue have already been discussed in educational literature [⁷, ⁸, ⁹].

In this paper, we study the electrical properties of an L-shaped conducting plate. Consideration of this problem is relevant for real printed circuit boards (PCB) and integrated circuits, where traces often have right angles. It is also important for resistance calculations in the case of thin film contacts [¹⁰, ¹¹, ¹²]. However, first of all, this work has an important pedagogical value, since it helps students understand the general principles of calculating the electrical characteristics of conducting objects with non-trivial geometry. The topic will be of interest and useful to undergraduate students studying advanced electrodynamics.

2. Theory

Let us consider an L-shaped conducting plate of constant thickness $t$ and uniform resistivity $ρ$ . Each of the two identical arms has length $L / 2$ and width $a$ $(a \leq L / 2)$ . A constant voltage $V$ is applied between the two ends of the plate, as shown in Fig. 1. Since the thickness is small compared to the in-plane dimensions, the problem can be treated as two-dimensional.

Figure 1
Geometry of the problem.

In the steady-state DC regime, charge accumulation inside the conductor is absent. Therefore, the electric potential $Φ (x, y)$ satisfies the Laplace equation

(1)

\nabla^{2} Φ = \frac{\partial^{2} Φ}{\partial x^{2}} + \frac{\partial^{2} Φ}{\partial y^{2}} = 0 .

The boundary conditions follow directly from the physical constraints imposed on the system. The two ends of the plate are maintained at fixed electric potentials, which leads to Dirichlet boundary conditions:

(2)

Φ (L / 2, y) = 0, 0 \leq y \leq a,

(3)

Φ (x, L / 2) = V, 0 \leq x \leq a .

All remaining boundaries are electrically insulating. This implies that no current flows through the lateral surfaces and, consequently, that the normal component of the electric field vanishes at these boundaries. Using the relation $E = - \nabla Φ$ , this condition can be written as the Neumann boundary condition

(4)

\frac{\partial Φ}{\partial n} = 0,

where $n$ is the outward normal vector to the boundary. Although the analysis is carried out in two dimensions, the results apply equally to a thick conducting plate with electrically insulated top and bottom surfaces. In this case, the Neumann boundary conditions enforce a uniform potential along the thickness, and the three-dimensional problem reduces exactly to the two-dimensional one.

Once the electric potential is known, the electric field inside the plate is obtained as

(5)

E = - \nabla Φ .

Its magnitude is given by

(6)

E = \sqrt{{(\frac{\partial Φ}{\partial x})}^{2} + {(\frac{\partial Φ}{\partial y})}^{2}} .

The current density is related to the electric field through Ohm’s law in differential form,

(7)

j = \frac{1}{ρ} E .

The total current flowing through the plate can be obtained by integrating the normal component of the current density over an arbitrary cross section connecting the two electrodes:

(8)

I = \int_{S} j \cdot d S .

In the steady-state DC regime, charge conservation implies $\nabla \cdot j = 0$ , and therefore the total current $I$ is independent of the particular choice of the cross section $S$ .

For convenience, we evaluate the current at the boundary $x = L / 2$ , where the electric potential is fixed and equal to zero. In this region, the equipotential lines are perpendicular to the boundary, and the electric field is strictly parallel to the $x$ -axis. Consequently, only the $x$ -component of the electric field contributes to the current.

Taking into account the constant thickness $t$ of the plate, the current reduces to

(9)

I = - {\frac{t}{ρ} \int_{0}^{a} \frac{\partial Φ}{\partial x} |}_{x = L / 2} d y .

Finally, the electrical resistance of the L-shaped plate is defined in the usual way as

(10)

R = \frac{V}{I} .

3. Results and Discussion

Since there is no closed analytical solution to the system of (equations 1)–(4), we solve it numerically. The numerical evaluation of the electric potential and field distribution in an L-shaped conducting plate was carried out using the finite element method (FEM) applied to the Laplace equation (1) with Dirichlet and Neumann boundary conditions. The computational domain was defined as a square with a cut-out and the mesh was adaptively refined near the re-entrant corner and along the boundaries with fixed potentials to ensure numerical stability and accuracy of the flux computation. A Delaunay triangulation was employed to generate the unstructured mesh, after which local stiffness matrices were assembled for each triangular element based on the gradients of linear basis functions.

The global sparse linear system was solved using the SciPy sparse solver, with an optional small diagonal regularization term added to prevent singularity of the stiffness matrix. The entire implementation was developed in Python [¹³], utilizing NumPy, SciPy, and Matplotlib libraries, providing flexibility in mesh resolution and full reproducibility of numerical experiments.

Although no closed-form solution in terms of a finite combination of elementary functions exists, there is a lengthy series solution involving sine and hyperbolic basis functions [¹¹].

The electric field distribution within L-shaped plate for different ratios $a / L$ is shown in Fig. 2. In Fig. 3 we plot the set of equipotential electric lines for the same cases. It follows from Fig. 3 that the highest values of the electric field strength correspond to the internal right angle of the plate, while the lowest values occur for the external right angle. Accordingly, current crowding occurs near the inner corner [¹⁴].

Figure 2
FEM simulation result for electric field in L-shaped conducting plate.

Figure 3
Set of equipotential electric lines within L-shaped conducting plate.

It is known [¹⁵] that the local behavior of solutions of the two-dimensional Laplace equation in the vicinity of a sharp corner is governed by a power-law singularity. In a planar angular domain of opening angle $α$ , the leading term of the electric potential scales as $Φ \sim r^{π / α}$ , where $r$ denotes the radial distance from the vertex of the corner. Consequently, the magnitude of the electric field behaves as $E \sim r^{π / α - 1}$ .

In the present problem, homogeneous Neumann boundary conditions (4) are imposed on the lateral edges end the internal (re-entrant) right angle corresponds to an opening angle $α = 3 π / 2$ , yielding $E \sim r^{- 1 / 3}$ and therefore a singular electric field near the inner corner. In contrast, the external right angle corresponds to $α = π / 2$ , for which $E \sim r$ , implying that the electric field vanishes as the corner is approached.

In real physical systems (for example, PCB traces), the divergence of the electric field can be regularized by rounding the inner corner. If the radius of curvature of the the inner corner becomes comparable to the plate thickness, the current redistribution becomes fully three-dimensional and the two-dimensional corner asymptotics no longer applies.

It is instructive to compare the above behavior with the electrostatic case of a charged conducting plate, for which Dirichlet boundary conditions apply. In that situation, the roles of the two corners are reversed: the external electric field diverges near the outer (convex) corner ( $α = 3 π / 2$ , $E \sim r^{- 1 / 3}$ ), while it vanishes near the inner (concave) corner ( $α = π / 2$ , $E \sim r$ ). This comparison highlights the fundamentally different physical origins of the field enhancement in steady-state current flow and in electrostatic equilibrium, despite the fact that both problems are governed by the Laplace equation.

Let us introduce the resistance of the straightened plate corresponding to a given L-shaped plate:

(11)

R_{0} = ρ \frac{L}{a t} .

The reference resistance $R_{0}$ corresponds to the value that would be obtained by treating the L-shaped plate as an equivalent straight strip of width $a$ and total length $L$ , equal to the sum of the two external arms. This choice does not represent an alternative physical geometry, but rather a naive one-dimensional estimate that neglects the two-dimensional redistribution of the current. In this sense, $R_{0}$ provides an upper-bound reference corresponding to the longest geometrical path between the terminals. This is the quantity that most students typically use to estimate the resistance of an L-shaped conducting plate. In any case, it is convenient to normalize the actual resistance value in numerical calculations to some reference value (for example, $R_{0}$ ).

It turns out that true resistance (10) is always less than the corresponding resistance (11) (Fig. 4). $R = R_{0}$ only in the limiting case $a / L \to 0$ . As $a / L$ increases, the ratio $R / R_{0}$ decreases monotonically (Fig. 4). The monotonic decrease of the normalized resistance $R / R_{0}$ with increasing aspect ratio $a / L$ can be understood in terms of the qualitative change in the current-flow pattern induced by the geometry. In a straight conductor, the electric field lines and current streamlines are parallel, and the resistance is well described by the classical expression $R = ρ L / (a t)$ , which assumes a uniform current density and a fixed conduction length.

Figure 4
Relative resistance

R / R_{0}

versus aspect ratio

a / L

.

In contrast, in the L-shaped plate the electric field lines are strongly curved, and the current is distributed over multiple non-parallel paths. This allows a fraction of the current to flow along trajectories that are effectively shorter than the total geometric length $L$ , while other regions act as parallel conduction channels. From a circuit perspective, the L-shaped geometry therefore behaves as a combination of parallel current paths rather than as a single one-dimensional resistor, which naturally leads to a reduction of the total resistance.

The limiting case $a / L \to 0.5$ deserves special attention. In this limit, the two terminals become adjacent rather than opposite, and the configuration no longer corresponds to a conventional two-terminal conductor. Although this situation is not physically realizable as a standard resistance measurement, it is nevertheless instructive from a conceptual point of view. The electric field lines in this limit are no longer approximately parallel anywhere in the domain, clearly indicating a qualitatively different current-flow regime. The pronounced departure from the assumptions underlying the classical resistance formula explains the strong reduction of $R / R_{0}$ observed in Fig. 4 and highlights why simple straight-conductor approximations fail for conductors with sharp bends and large transverse dimensions.

4. Didactical Implications and Use in Undergraduate Courses

The numerical problem discussed in this work can be naturally formulated as a guided computational activity for undergraduate students with basic knowledge of electromagnetism and numerical methods. The activity may be structured as follows.

First, students download the provided Python code [¹⁶] and familiarize themselves with its overall structure. The code is organized in a modular way, separating mesh generation, finite-element assembly, solution of the Laplace equation, flux integration, and visualization. Students are encouraged to identify the role of the main classes (LShapedMesh, FEMSolver, FluxIntegrator, and ResistanceCalculator) and to trace the data flow from the geometric parameters to the final resistance value.

As a preliminary task, students examine the geometric parameters of the problem, in particular the ratio $a / L$ , and relate them to the shape of the L-shaped domain. By modifying the mesh parameters (e.g., base resolution and local refinement near the re-entrant corner), they can study the influence of mesh density on the stability and convergence of the computed resistance.

In a second step, students run the code for selected values of $a / (2 l)$ and visualize the resulting electric potential distributions. By inspecting the equipotential lines, they can directly observe the distortion of the electric field near the inner corner and compare it with the nearly uniform field far from the corner. This visual analysis helps connect the numerical results with the qualitative concepts of current crowding and field singularities.

The next task consists in computing the dimensionless resistance $R / R_{0}$ as a function of the geometric parameter $a / L$ . Students generate the resistance curve and compare it with the value predicted by the uniform-resistor formula. They are asked to identify the parameter range in which the standard formula fails and to relate this failure to the nonuniform topology of the current lines.

As an advanced exercise, students may modify the code to evaluate the current flux through different cross-sections of the conducting plate and verify the conservation of current numerically. They may also test alternative boundary conditions or extend the code to different geometries, such as U-shaped or rectangular domains, thereby exploring the generality of the numerical approach.

This computational activity allows students to simultaneously deepen their understanding of electrostatics, steady-state current flow, and numerical solutions of partial differential equations, while gaining hands-on experience with finite-element techniques in a physically cumbersome geometry.

It should be emphasized that the problem considered in this work is inherently counter-intuitive from a pedagogical point of view. In particular, students often tend to confuse the present steady-state current-flow problem with a purely electrostatic one, implicitly assuming that the electric field must necessarily exhibit a singularity at the external right angle of the plate. This expectation, however, is rooted in electrostatic intuition and does not apply under the boundary conditions relevant to current conduction. In addition, students frequently assume that the resistance of the plate can be evaluated using the standard uniform-resistor equation (11), overlooking the fact that this expression relies on an effectively one-dimensional current distribution. The L-shaped geometry discussed here provides a clear example of how such intuitive assumptions fail, thereby highlighting the need for a careful distinction between electrostatic and conductive boundary-value problems.

5. Conclusions

In this work, the electrical potential and field distribution in an L-shaped conducting plate were analyzed using the finite element method. The results demonstrate that the electric field exhibits a pronounced singularity near the inner corner of the plate, where current crowding occurs, while the outer corner corresponds to a region of reduced field intensity. The normalized electrical resistance of the L-shaped plate was found to decrease monotonically with increasing aspect ratio $a / L$ , remaining lower than that of the equivalent straightened configuration.

The L-shaped conducting plate provides a surprisingly rich example for undergraduate physics courses. Although the governing equation is simply Laplace equation, the geometry introduces effects that are rarely discussed in standard textbooks but are essential for building correct physical intuition about fields and currents in real devices. The system is ideal for student projects because the geometry is simple and easy to mesh and numerical solutions can be obtained with elementary finite-element codes.

From a didactic perspective, this application allows students to grasp several key concepts, such as: non-homogeneous electric field, field singularities, model approach, current density, finite element method. Finally, this topic contains a counter-intuitive effect, that are little explored in the text-books.

References

^[1] B.D. Popović, Introductory Engineering Electromagnetics (Addison-Wesley, Reading, 1971), 2 ed.
^[2] J.S. Huang, E.C.C. Yeh, Z.B. Zhang and K.N. Tu, Mater. Chem. Phys. 77, 377 (2002).
^[3] K. Luewattana, P. Rattanasena, IJECE 11, 4689 (2021).
^[4] I.M. Blankson, J. Electrost. 16, 33 (1984).
^[5] M. Elliotis, G. Georgiou and C. Xenophontos, Commun. Numer. Meth. Engng. 18, 213 (2002).
^[6] Y. Ai, S. Park, J. Zhu, X. Xuan, A. Beskok and S. Qian, Langmuir. 26, 2937 (2009).
^[7] J.D. Romano and R.H. Price, Am. J. Phys. 64, 1150 (1996).
^[8] V. Ivchenko, Rev. Bras. Ens. Fis. 42, e20200189 (2020).
^[9] E.J. Tan and B. Ricardo, The Physics Educator. 7, 2550010 (2025).
^[10] P.M. Hall, Thin Solid Films. 1, 277 (1968).
^[11] P. Zhang, Y.Y. Lau and R.M. Gilgenbach, J. Appl. Phys. 109, 124910 (2011).
^[12] P. Zhang, D.M.H. Hung and Y.Y. Lau, J. Phys. D: Appl. Phys. 46, 065502 (2013).
^[13] E.I.B. Rodrigues, M.L.S. Dias, R.O. Domingues, Rev. Bras. Ens. Fis. 47, e20240422 (2025).
^[14] P. Yang, S. Banerjee, W. Kuang, Y. Ding, Q. Ma and P. Zhang, J. Phys. D: Appl. Phys. 53, 485303 (2020).
^[15] W.A. Strauss, Partial Differential Equations: An Introduction (John Wiley and Sons, Hoboken, 2007), 2 ed.
^[16] V. Ivchenko, Electrical properties of an L-shaped conducting plate, available in: <ref class="ltx_nolink ltx_url" href="https://zenodo.org/records/18259525">https://zenodo.org/records/18259525</ref>
» https://zenodo.org/records/18259525

Edited by

Editor-in-Chief:
Marcello Ferreira https://orcid.org/0000-0003-4945-3169

Publication Dates

Publication in this collection
15 May 2026
Date of issue
2026

History

Received
24 Nov 2025
Reviewed
19 Mar 2026
Accepted
20 Mar 2026

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[1] ^[1] B.D. Popović, Introductory Engineering Electromagnetics (Addison-Wesley, Reading, 1971), 2 ed.

[2] ^[2] J.S. Huang, E.C.C. Yeh, Z.B. Zhang and K.N. Tu, Mater. Chem. Phys. 77, 377 (2002).

[3] ^[3] K. Luewattana, P. Rattanasena, IJECE 11, 4689 (2021).

[4] ^[4] I.M. Blankson, J. Electrost. 16, 33 (1984).

[5] ^[5] M. Elliotis, G. Georgiou and C. Xenophontos, Commun. Numer. Meth. Engng. 18, 213 (2002).

[6] ^[6] Y. Ai, S. Park, J. Zhu, X. Xuan, A. Beskok and S. Qian, Langmuir. 26, 2937 (2009).

[7] ^[7] J.D. Romano and R.H. Price, Am. J. Phys. 64, 1150 (1996).

[8] ^[8] V. Ivchenko, Rev. Bras. Ens. Fis. 42, e20200189 (2020).

[9] ^[9] E.J. Tan and B. Ricardo, The Physics Educator. 7, 2550010 (2025).

[10] ^[10] P.M. Hall, Thin Solid Films. 1, 277 (1968).

[11] ^[11] P. Zhang, Y.Y. Lau and R.M. Gilgenbach, J. Appl. Phys. 109, 124910 (2011).

[12] ^[12] P. Zhang, D.M.H. Hung and Y.Y. Lau, J. Phys. D: Appl. Phys. 46, 065502 (2013).

[13] ^[13] E.I.B. Rodrigues, M.L.S. Dias, R.O. Domingues, Rev. Bras. Ens. Fis. 47, e20240422 (2025).

[14] ^[14] P. Yang, S. Banerjee, W. Kuang, Y. Ding, Q. Ma and P. Zhang, J. Phys. D: Appl. Phys. 53, 485303 (2020).

[15] ^[15] W.A. Strauss, Partial Differential Equations: An Introduction (John Wiley and Sons, Hoboken, 2007), 2 ed.

[16] ^[16] V. Ivchenko, Electrical properties of an L-shaped conducting plate, available in: <ref class="ltx_nolink ltx_url" href="https://zenodo.org/records/18259525">https://zenodo.org/records/18259525</ref>
» https://zenodo.org/records/18259525