What the equal sign means in physics

[Dr. Courtney]

For example F=ma means that the definition of force is m*a or the quantity of left side equals to the quantity of right side or both ? or kinetic energy..we know K=1/2mu^2 but is this the definition of kinetic energy or just the formula to calculate it ?

The meaning of the equal sign in physics depends a lot on context, and the matter is not without its subtleties. For equations in physics to have physical meaning, you always need to be aware of the fine print which gives more explanation including:

- The definitions of the symbols in the equation
- The units of the physical quantities represented by the symbols
- Special conditions required for the equation to be exactly true
- Special conditions required for the equation to be true to an excellent approximation

It is rare that all the needed explanation is given along with an equation. But some of the explanations needed for F = ma are:

- F is the force in Newtons, m is the mass in kg, and a is the acceleration in m/s/s (or another consistent system of units
- The motion and forces are confined to one dimension (otherwise a vector equation is needed)
- Either the force is the net external force or only one external force is acting
- The mass of the object is not changing
- The acceleration refers to the center of mass of the object
- Quantum and relative effects may be neglected

No, but you can use it as definition (because of the equality) for the force needed to accelerate a mass, or to stop one. However, this might not help you to understand force, as it only applies to this one case. What is the force needed to stretch a spring? Or the force between two magnets? But according to Newton's laws of motion, by which force is defined by a change of the velocity of a mass, this definition will do in may cases.

Since the relevant F in F = ma is a net force, F = ma is only useful as a definition of force if we can be sure only one force is acting on the object. In practice, this may be a useful approximation, but without an outside understanding of what forces are, one quickly runs into a circular argument supporting the case that additional forces are not acting. In practice, F = ma is more useful as the definition of intertial mass. Inertial mass is the constant of proportionality between an applied net force and the resulting acceleration.

 

[Stephen Tashi]

In mathematics, the equal sign means that two items are the same in the algebra that has been defined,

Yes. For example "=" for real numbers isn't the same relation as "=" for matrices.

 

[fresh_42]

Yes. For example "=" for real numbers isn't the same relation as "=" for matrices.

One can even discuss whether $1=\frac{2}{2}$ is a sign for "the same" or "the equivalent". It becomes more obvious if we have examples like $[0]=[2] \in \mathbb{Z}/2\mathbb{Z}\,.$

 

[Chandra Prayaga]

That relation does not apply to the static E field inside of the battery, or any other source of emf. Unlike in a resistor where the current does move in the direction of the static E field, and your relation applies.

The specific relation between electric field and current should not matter in this case. If the net electric field inside the battery is zero, then there is no force on the charge carriers, and no current inside the battery.

 

[jbriggs444]

The specific relation between electric field and current should not matter in this case. If the net electric field inside the battery is zero, then there is no force on the charge carriers, and no current inside the battery.

If the net electric field is zero there is no force on the charge carriers, no change in their average velocities and no change in the associated current.

The current through a circuit with zero electric field everywhere is indeterminate.