Law: free energy change in dielectrics

symplyphysics/laws/thermodynamics/dielectrics/free_energy_change_via_temperature_change_and_electric_field_change.py

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+"""
+Free energy change via temperature change and electric displacement change
+==========================================================================
+
+The infinitesimal change in (Helmholtz) free energy of a system with a dielectric
+medium can be expressed using the change in temperature and the change in electric
+displacement.
+
+**Conditions:**
+
+#. The dielectric is isotropic whether or not the electric field is present.
+#. The medium is homogeneous.
+#. The volume change of the medium is insignificant.
+
+**Links:**
+
+#. Formula 31.7 on p. 122 of "General Course of Physics" (Obschiy kurs fiziki), vol. 3 by Sivukhin D.V. (1979).
+"""
+
+from sympy import Eq
+from symplyphysics import (
+    symbols,
+    units,
+    Quantity,
+    SymbolNew,
+    clone_as_symbol,
+    validate_input,
+    validate_output,
+)
+
+free_energy_density_change = SymbolNew("dH", units.energy / units.volume)
+"""
+Infinitesimal change in :symbols:`helmholtz_free_energy` of the system
+per unit :symbols:`volume`.
+"""
+
+entropy_density = SymbolNew("S", units.energy / units.temperature / units.volume)
+"""
+:symbols:`entropy` of the system per unit :symbols:`volume`.
+"""
+
+temperature_change = clone_as_symbol(
+    symbols.temperature,
+    display_symbol="dT",
+    display_latex="dT",
+)
+"""
+Infinitesimal change in :symbols:`temperature` of the system.
+"""
+
+electric_field_strength = symbols.electric_field_strength
+"""
+:symbols:`electric_field_strength`.
+"""
+
+electric_displacement_change = clone_as_symbol(
+    symbols.electric_displacement,
+    display_symbol="dD",
+    display_latex="dD",
+)
+"""
+Infinitesimal change in :symbols:`electric_displacement` of the system.
+"""
+
+law = Eq(
+    free_energy_density_change,
+    -1 * entropy_density * temperature_change + electric_field_strength * electric_displacement_change,
+)
+"""
+:laws:symbol::
+
+:laws:latex::
+"""
+
+# Derivation
+## Make use of the formula for free energy and the first law of thermodynamics applied to dielectric media.
+
+
+@validate_input(
+    entropy_density_=entropy_density,
+    temperature_change_=temperature_change,
+    electric_field_strength_=electric_field_strength,
+    electric_displacement_change_=electric_displacement_change,
+)
+@validate_output(free_energy_density_change)
+def calculate_free_energy_density_change(
+    entropy_density_: Quantity,
+    temperature_change_: Quantity,
+    electric_field_strength_: Quantity,
+    electric_displacement_change_: Quantity,
+) -> Quantity:
+    result = law.rhs.subs({
+        entropy_density: entropy_density_,
+        temperature_change: temperature_change_,
+        electric_field_strength: electric_field_strength_,
+        electric_displacement_change: electric_displacement_change_,
+    })
+    return Quantity(result)

test/thermodynamics/dielectrics/free_energy_change_via_temperature_change_and_electric_field_change_test.py

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+from collections import namedtuple
+from pytest import fixture, raises
+from symplyphysics import (
+    assert_equal,
+    errors,
+    units,
+    Quantity,
+)
+from symplyphysics.laws.thermodynamics.dielectrics import (
+    free_energy_change_via_temperature_change_and_electric_field_change as law,
+)
+
+Args = namedtuple("Args", "s dt e dd")
+
+
+@fixture(name="test_args")
+def test_args_fixture() -> Args:
+    s = Quantity(10 * units.joule / units.kelvin / units.meter**3)
+    dt = Quantity(-1e-3 * units.kelvin)
+    e = Quantity(15 * units.volt / units.meter)
+    dd = Quantity(3e-3 * units.coulomb / units.meter**2)
+    return Args(s=s, dt=dt, e=e, dd=dd)
+
+
+def test_law(test_args: Args) -> None:
+    result = law.calculate_free_energy_density_change(test_args.s, test_args.dt, test_args.e, test_args.dd)
+    assert_equal(result, 5.5e-2 * units.joule / units.meter**3)
+
+
+def test_bad_entropy_density(test_args: Args) -> None:
+    sb = Quantity(units.candela)
+    with raises(errors.UnitsError):
+        law.calculate_free_energy_density_change(sb, test_args.dt, test_args.e, test_args.dd)
+    with raises(TypeError):
+        law.calculate_free_energy_density_change(100, test_args.dt, test_args.e, test_args.dd)
+
+
+def test_bad_temperature(test_args: Args) -> None:
+    tb = Quantity(units.candela)
+    with raises(errors.UnitsError):
+        law.calculate_free_energy_density_change(test_args.s, tb, test_args.e, test_args.dd)
+    with raises(TypeError):
+        law.calculate_free_energy_density_change(test_args.s, 100, test_args.e, test_args.dd)
+
+
+def test_bad_electric_field(test_args: Args) -> None:
+    eb = Quantity(units.candela)
+    with raises(errors.UnitsError):
+        law.calculate_free_energy_density_change(test_args.s, test_args.dt, eb, test_args.dd)
+    with raises(TypeError):
+        law.calculate_free_energy_density_change(test_args.s, test_args.dt, 100, test_args.dd)
+
+
+def test_bad_electric_displacement(test_args: Args) -> None:
+    db = Quantity(units.candela)
+    with raises(errors.UnitsError):
+        law.calculate_free_energy_density_change(test_args.s, test_args.dt, test_args.e, db)
+    with raises(TypeError):
+        law.calculate_free_energy_density_change(test_args.s, test_args.dt, test_args.e, 100)