electromagnetism
Why are there two rules: Fleming's left hand and right hand rules? What is the difference between the two and why can't we use just one rule?
Suppose the magnetic field is from right to left and the motion of the wire is downwards then according to the right hand rule the induced current will be in the straight direction.
But if we use the left hand rule in this same situation to find the direction of motion of wire then it shows that the direction of wire is upwards.
Please help me.
Note that a current carrying wire produces a circular magnetic field that's why it doesn't matter how you hold your hand ie how you rotate your hand around your arm as long as your thumb shows the direction of the current.
Edit after comments:
See the illustration I've added below. Now use your hand in the way that you've learned and convince yourself that what I've drawn below is correct.
You could just calculate the vector components. These hand rules are used whenever vector quantities are related by a cross product. Let's take as an example the force on a moving charge due to a magnetic field: $$\vec{F}=q \vec{v} \times \vec{B}$$
Let's write this out in terms of its components (assuming we're working in 3D):
$$\left( \begin{array}{} F_x \\ F_y \\ F_z \end{array} \right) = q \left( \begin{array}{} v_x \\ v_y \\ v_z \end{array} \right) \times \left( \begin{array}{} B_x \\ B_y \\ B_z \end{array} \right) = q \left( \begin{array}{} v_y B_z - v_z B_y \\ v_z B_x - v_x B_z \\ v_x B_y - v_y B_x \end{array} \right)$$
In principle, this gives you the direction of the force... but I doubt it will be faster than using a hand rule. I generally only use one hand rule, namely the right hand rule for cross products:
Given two vectors $\vec{A}$ and $\vec{B}$, point the fingers of your right hand in the direction of $\vec{A}$, then curl them towards $\vec{B}$. Your thumb then points in the direction of the outer product $\vec{A}\times\vec{B}$.
If you know the equation for the cross product quantity you're trying to calculate, you can always use this rule and don't have to think about whether you need to use a left-hand or right-hand rule.
Best Answer
It is unfortunate that the physics of magnetism got saddled with several different *-hand rules, and that they use different hands. Let's pull them apart:
Fleming's left-hand rule
gives you the direction of the force that acts on a current if you know the magnetic field.
Image source
This rule applies to motors, i.e. devices which use currents in a magnetic field to generate motion. It derives its validity from the Lorentz force, $$ \mathbf F=q\mathbf v\times\mathbf B, $$ in which the current goes with the charge's velocity and the induced motion is along the direction of the force. This is why this rule coincides with the left-hand rule used in cross-products in general.
Fleming's right-hand rule
is much less used in physics (though I can't speak for how engineers do things). It applies to generators, i.e. devices which use motion in a magnetic field to generate currents. This again relies on the cross product in the Lorentz force, except that now the charge's velocity is given by the object's motion, and the force along the wire is what establishes the current. This means you've swapped the middle finger with the thumb with respect to Fleming's left-hand rule, which you can do by keeping the (vague) assignments to 'motion' and 'current' and switching hands.
Image source
I dislike this convention very much and I would encourage you to forget all about it except the fact that it exists and should be avoided. In any situation where you need it, you can simply use the Lorentz force to figure out which way the current will go.
Ampère's right-hand rule
is quite different, and it gives you the magnetic field generated by a straight wire.
Image source
It derives its validity from the Biot-Savart law, which gives the magnetic field at position $\mathbf r$ generated by an infinitesimal current element of current $I$ and directed length $\mathrm d\mathbf l$ at position $\mathbf r'$, as $$ \mathbf B(\mathbf r)=\frac{\mu_0}{4\pi}\frac{I\mathrm d\mathbf l\times(\mathbf r-\mathbf r')}{|\mathbf r-\mathbf r'|^3} $$ Again, it is the cross product which dictates the direction of the field, and you should check by yourself that it works out as indicated in the picture.
As you can see, the rules are quite different. It is therefore crucial that, if you want to use them as mnemonics, you learn correctly which one applies where, and that you apply them correctly. (It is no use to learn which hand to use if you e.g. swap the assignments for the index and middle finger.)
The most important thing to learn, though, is the Lorentz force law, which is based on a left-hand rule (charge-times-current on your middle finger, field on the index, force on the thumb) indicated by the cross product. This is essentially failsafe if you apply it correctly and is less subject to confusion with other rules.