f(x); g(x)

Where is a comparison operator that can be greater than (>), less than (<), greater than or equal to (≥), less than or equal to (≤), or different from (≠), and "f(x)" and "g(x)" are any two functions of a variable x.

This formula shows that the comparison operator is used to compare the functions f(x) and g(x), allowing for the establishment of an order relationship between the two functions. The values of x that comply with the ordering relation provided by the comparison operator are referred to as the solution of the inequality.

In particular, in the field of physics and the study of electrical circuits, inequalities allow determining minimum or maximum values for electric current or electric charge, which is crucial to ensure the proper operation of circuits and the safety of the people and related equipment.

Inequalities are also necessary for the analysis and design of more sophisticated electrical systems, such as power distribution networks, in which resource optimization is necessary to meet consumer demands and ensure the stability and security of the system. The ability to set limitations and restrictions on the variables involved to ensure the proper operation of the systems and the safety of people and equipment makes inequalities an effective tool for solving quantitative problems in physics and electrical charges.

Although they are not specifically related to electric charges, equations are frequently used in mathematics to solve issues of inequality and can be helpful in instances when you need to determine the quantity of electric charge required to accomplish a particular goal.

An inequality, for instance, can be used to determine how much charge is needed to reach a given voltage in an electrical circuit. Let's say a circuit has a power supply of 12 volts and a resistance of 10 ohms. You can use the inequality to determine how much electrical charge is required in the circuit to get a voltage of 10 volts:

Q/V = I ≤ V/R

Q is the electric charge expressed in coulombs, V the voltage expressed in volts, I the current expressed in amps, and R the resistance expressed in ohms. According to the inequality, the voltage divided by the resistance must be less than or equal to the electric current.

When the inequality for the electric charge Q is solved, we arrive at:

Q ≤ V*I

With the known values substituted, we obtain:

Q ≤ (10V)(1A) = 10C

Hence, 10 coulombs of electrical charge are needed in this circuit to achieve a voltage of 10 volts.

It is crucial to keep in mind that, in reality, electrical loads can be more complicated than in this simplified example, and that further calculations could be necessary to take things like capacitance, inductance, and other electrical characteristics of the circuit into account. A useful mathematical tool for resolving issues with electric charge may also use inequalities.

Of course, here is an example of an inequality exercise utilizing electric charges:

The resistance of an electrical circuit is 20 ohms, and the power source is 24 volts. If the electrical current in the circuit is desired to be larger than or equal to 1.5 amps, what is the minimum quantity of electrical charge required?

To solve this problem, we can use the formula:

I = Q/t

Where I is the electric current in amperes, Q is the electric charge in coulombs, and t is the time in seconds.

We can clear the electric charge Q from the previous formula:

Q = I * t

Now, we can use Ohm's law to determine the circuit's equivalent resistance:

R = V/I

Where R is resistance in ohms, V is voltage in volts, and I is current in amps.

With the known values substituted, we obtain:

R equals 24 V x 1.5 A x 16 ohms.

Since the resistance of the circuit is 20 ohms, we can calculate the electric current that will flow in the circuit using Ohm's law:

I = V / R

With the known values substituted, we obtain:

I equals 24 V/20 Ohms, or 1.2 A.

We can write the inequality as follows since the electric current must be greater than or equal to 1.5 amps:

I ≥ 1.5 A

When we enter the equation for electric current into the inequality, we obtain:

Q/t ≥ 1.5 A

When we remove the inequality and solve for the electric charge Q, we get:

Q≥1.5A*t

Now we can change the formula for electric current into the equivalent resistance of the circuit to get:

I = V / (R + r)

When the generator's internal resistance (r) is concerned.

When we replace the known values, we obtain:

1.5 A = 24 V / (20 ohms + r)

Solving for r, we get:

r = (24V / 1.5A) - 20 ohms = 8 ohms

Using the electric charge formula, we can now determine how long it will take for the required electric current to flow:

Q≥1.5A*t

With the known values substituted, we obtain:

Q >= (1.5 A) * (t seconds) (t seconds)

After removing the time t, we get:

t ≤ Q / (1.5 A) (1.5 A)

When we add the generator's internal resistance to the equation for electric current, we obtain:

I = 24 V/(20 + 8) ohms = 1 A

Now we can determine the bare minimum electrical charge required:

Q ≥ (1.5 A) * (1 second) = 1.5 C

Therefore, the minimum amount of electrical charge needed is 1.5 coulombs to make the electrical current in the circuit greater than or equal to 1.5 amps, assuming a 20 ohm resistor and a 24 ohm power supply. volts with an internal resistance of 8 ohms.

Photo edited by my Samsung A23 phone

Bibliography Reference

Electronics Practices by Paul B. Zbar, Albert Paul Malvino, Michael A. Miller, 2003.

Physics for science and technology. II by Paul Allen Tipler, Gene Mosca, 2004.

Inequalities in Physics and Astrophysics by Michał Eckstein, 2018.