To tackle your question about the electric field inside a resistive ring placed in a coaxial solenoid with a time-varying magnetic field, we need to break down the concepts involved, including electromagnetic induction, Ohm's law, and Maxwell's equations. Let's dive into it step by step.
Understanding the Setup
We have a coaxial solenoid that generates a magnetic field described by the equation B = B₀t, where B₀ is a constant and t is time. This indicates that the magnetic field inside the solenoid is increasing linearly with time. When a resistive ring is placed within this solenoid, it experiences a changing magnetic field, which induces an electric field according to Faraday's law of electromagnetic induction.
Induced Electric Field
According to Faraday's law, the induced electromotive force (EMF) in the ring is given by:
where Φ is the magnetic flux through the ring. The magnetic flux can be expressed as:
where A is the area of the ring. For a ring of radius r, the area is A = πr². Thus, the flux becomes:
Taking the time derivative gives:
Substituting this into the EMF equation results in:
This induced EMF generates an electric field E around the ring, which can be related to the EMF by the equation:
From this, we can find the electric field:
Ohm's Law and the Electric Field
Now, regarding your question about Ohm's law, which states that E = ρj, where ρ is the resistivity and j is the current density. The electric field we calculated is indeed a part of the net electric field in the ring. The current density j can be expressed as:
where I is the current flowing through the ring and A is the cross-sectional area. The direction of the induced electric field follows the right-hand rule, which means it will circulate around the ring in a direction that opposes the change in magnetic flux (Lenz's law).
Maxwell's Third Equation
Maxwell's third equation, expressed as ∇ × E = -∂B/∂t, indicates that a changing magnetic field induces a curl in the electric field. In our case, since the magnetic field is changing with time, the electric field will have a non-zero curl. This means that the electric field is not uniform and has a rotational component due to the changing magnetic field.
Even though Ohm's law applies, the electric field induced by the changing magnetic field is separate from the resistive effects described by Ohm's law. The total electric field in the ring can be considered as a superposition of the induced electric field and any field due to resistive effects.
Summary
In summary, the electric field inside the resistive ring is given by E = -B₀r/2, and it circulates in a direction that opposes the change in magnetic flux. Ohm's law applies to the current density within the ring, and Maxwell's third equation confirms that the electric field has a curl due to the changing magnetic field. This interplay of induced electric fields and resistive effects is a fundamental aspect of electromagnetism.
, radius is r? Does E=rho times j , ie the Ohms law field a part of the net field? If so, then how can its direction be found? Also what about Maxwell's third equation- del cross E= minus the rate of change of magnetic field with time, if the Ohms law field still applies?
