- #1

AmanWithoutAscarf

- 22

- 1

- Homework Statement
- I cannot differentiate between the torque of Lorentz force and the one caused by induced electric field, in an electrodynamics problem of a charged ring falling in non-uniform magnetic field.

- Relevant Equations
- ##F_{B} =q\mathbf{v} \times \mathbf{B}##

## \nabla \times \mathbf{E} =-\frac{\partial \mathbf{B}}{\partial t}##

## \iint \frac{\partial \mathbf{B}}{\partial t} .dS=\oint \mathbf{E} .dl##

As far as I know, there are two causes of the rotational motion of the charged ring. T

he first torque is from Lorentz force: $$T=Q\mathbf{R\times ( v} \times \mathbf{B}_{r}) =-\frac{QR^{2}}{2}\frac{dz}{dt}\frac{d\mathbf{B}_{z}}{dz}$$ (where ##\mathbf{B}_{r} =-\frac{R}{2}\frac{d\mathbf{B}_{z}}{dz}## is the radial component of the magnetic field).

The second torque is derived from force of induced electric field: $$ F=E_{d} .Q=\frac{-d\upPhi }{dt} .\frac{1}{2\pi R} .Q=-\frac{R}{2}\frac{dB_{z}}{dt} .Q\ \Longrightarrow \ T_{1} =-\frac{QR^{2}}{2}\frac{d\mathbf{B}_{z}}{dt}$$

But it turns out that the two torque T and T1 are the same. And in the solution, only one torque was considered.

So is the Lorentz force-field the same as induced electric field?

(The picture is from https://www.numerade.com/ask/questi...nd-falls-to-the-ground-uhrough-a-non-u-67393/)

Here are all the aformentioned formulas, sorry for all language inconveniences T.T

he first torque is from Lorentz force: $$T=Q\mathbf{R\times ( v} \times \mathbf{B}_{r}) =-\frac{QR^{2}}{2}\frac{dz}{dt}\frac{d\mathbf{B}_{z}}{dz}$$ (where ##\mathbf{B}_{r} =-\frac{R}{2}\frac{d\mathbf{B}_{z}}{dz}## is the radial component of the magnetic field).

The second torque is derived from force of induced electric field: $$ F=E_{d} .Q=\frac{-d\upPhi }{dt} .\frac{1}{2\pi R} .Q=-\frac{R}{2}\frac{dB_{z}}{dt} .Q\ \Longrightarrow \ T_{1} =-\frac{QR^{2}}{2}\frac{d\mathbf{B}_{z}}{dt}$$

But it turns out that the two torque T and T1 are the same. And in the solution, only one torque was considered.

So is the Lorentz force-field the same as induced electric field?

(The picture is from https://www.numerade.com/ask/questi...nd-falls-to-the-ground-uhrough-a-non-u-67393/)

Here are all the aformentioned formulas, sorry for all language inconveniences T.T