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Chen et al. Soft Sci. 2026, 6, 9 Page 13 of 36

Lorentz-force manipulation
The pure LMs without magnetic agents can also be magnetically manipulated via the Lorentz force, which is
generated by the interaction between an external magnetic field and the moving electrons within the
conductive LM. The Lorentz force describes the total force F exerted on a point charge moving with velocity
v in a magnetic field B . It is given by:
[97]

F = Q · v × B
where F is the force vector, Q is the electric charge, v is the velocity vector, B is the magnetic field vector, and
v × B denotes the cross product of the velocity and magnetic field vectors.

Moreover, the force acting on each moving charge can be extended to describe the macroscopic force exerted
on a current-carrying conductor . For a wire segment with current I, length vector L, and uniform magnetic
[98]
field B, the force becomes:

F = I (L × B)
L
This expression follows from aggregating the Lorentz force over all moving charges within the conductor,
where the drift velocity of charges and the linear charge density are incorporated into the definition of the
electric current I:

I = nqAvd
where n is the charge carrier density, q is the charge per charge carrier, A is the cross-sectional area, and v is
d
the drift velocity.

Electrical currents are primarily categorized by their origin into applied and induced currents. Applied
current, also known as conduction current, is generated by an external power source such as a battery or
generator, which establishes an electric field to drive charge carriers through a conductive path. In contrast,
an induced current is created through electromagnetic induction. A changing magnetic field generates a
circulating electric field, which in turn causes electrons to flow within a conductor.

Applied current
For optimal current distribution and magnetic manipulation, applied currents are typically introduced to the
LM wires, which can be fabricated by injecting LM into the flexible microfluidic channels . The relationship
[98]
between voltage, current, and resistance in the LM wire is described by Ohm’s law. For a homogeneous
conductor at constant temperature, the current density J can be related to the electric field E through the
microscopic form of Ohm’s law:

J = σE
where σ is the electrical conductivity of the material. This microscopic formulation can be integrated to yield
the macroscopic form:

V = IR
where V is the voltage across the conductor, I is the current flowing through it, and R is its electrical
resistance. The resistance R is related to the material’s conductivity σ and the conductor’s geometry through:

R = L/(σA)
where L is the length of the conductor and A is its cross-sectional area.