homework and exercises - Lenz' law versus $-frac{dOmega}{dt}$

I am preparing for an exam, on one problem my answer differ from the solution.

The current $i(t)=I_0e^{-\alpha t}$ runs in a long straight conductor along $\hat z$.

Point A,B,C,D forms a triangle. A and B is at $L \hat x$ with a very small separation in $\hat z$ with a resistance R between. C and D is at $2L \hat x$ with a separation in $\hat z$ of $2L$.

The question is: What is the induced current in the triangle.

I calculated the flux $\Omega$ correctly, but my answer differ on the direction of the current.

I took $V_{ind}=-\frac{d\Omega}{dt}$, since $i(t)$ is decreasing, the magnetic flux passing through ABCDA is decreasing, and so (to my understanding) it is reasonable that $i_{ABCD}=\frac{V_{ind}}{R}$ will run clockwise (using right-hand notation).

However, in the solution to the problem they too use $V_{ind}=-\frac{d\Omega}{dt}$ but then "magically" on the next line remove the minus sign and says that according to Lenz law the current opposes the magnetic flux that caused the current and so will run counter-clockwise.

I would be thankful for an intuitive understanding for why it is opposing the magnetic flux that caused the current and not opposing the change in the magnetic flux? Thanks.

Answer

Actually, it does oppose the change in the magnetic flux causing it. If the magnetic field from the source is increasing in some direction, the magnetic field from the induced current decreases in the same direction to oppose the increase, for example. The induced current opposing the change causing it ensures that the cause must supply energy to the system to increase the induced current, so guaranteeing conservation of energy.

The magnetic field from the wire points towards $\hat y$ and is decreasing, which means the magnetic field has to increase in the same direction from the induced current. Therefore the current runs clockwise around the loop.

The negative sign in Lenz's law is needed when using the right hand rule to find the direction of the induced emf: The thumb points in the direction of the applied magnetic field, and the curled fingers point around the direction of the induced emf. In your example, the thumb points towards $\hat y$ and the fingers curl in a clockwise sense. The magnetic field decreasing cancles the minus sign so the direction is the same as the fingers - clockwise.