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A magnetic field is an invisible force that is created by electric currents and magnetic materials. It is a vector field, meaning it has both a direction and a magnitude. The magnetic field plays a crucial role in the working of a DC motor. In this context, it influences the force exerted on a current-carrying coil, which in turn affects the torque produced by the motor.

When a coil carrying current is placed in a magnetic field, it experiences a force due to the magnetic field. This is because of the interaction between the magnetic field and the electric current, which causes the coil to rotate. The strength of the magnetic field is measured in Teslas (T). This force is maximized when the angle between the magnetic field lines and the normal to the coil is 90 degrees.

In our problem, a magnetic field of 0.20 T is used to help calculate the torque and ultimately find the current in the motor. Understanding magnetic fields helps predict how forces will act in different configurations, which is essential for designing efficient motors.

Torque is a measure of the rotational force on an object like a coil. It tells us how much force is needed to rotate an object around an axis. In a DC motor, the torque is generated when the current-carrying coil interacts with the magnetic field. The formula for calculating the torque (\(\tau\)) is given by:
\[t\tau = n \cdot I \cdot A \cdot B \cdot \sin(\theta) \]
- **\(n\)**: the number of turns in the coil (1200 turns in our problem)- **\(I\)**: the current through the coil (the value we need to find)- **\(A\)**: the area of a single turn of the coil- **\(B\)**: the strength of the magnetic field- **\(\theta\)**: the angle between the magnetic field and the area vector of the coil, which is 90 degrees when torque is maximum

For maximum torque, \(\sin(\theta)\) is equal to 1 because \(\theta = 90^{\circ}\). This simplifies our calculations. We substitute the known values into the formula and solve for \(I\). This allows us to determine how much current is required to produce a given amount of rotational force.

Electric current is the flow of electric charge and is measured in Amperes (A). In DC motors, electric current flows through the coils, creating a magnetic field that interacts with an external magnetic field, producing torque and causing motion.

In the given exercise, finding the electric current is crucial to understanding how the motor produces the desired torque. By using the calculated formula derived from torque calculation, \(I = \frac{\tau}{n \cdot A \cdot B}\), students can determine the current that ensures the motor operates optimally. This involves rearranging the formula to solve for the current and involves substituting all the known quantities related to torque and the magnetic field.

Knowing the amount of electric current needed helps in practical applications, like setting the right voltage or checking energy consumption in real-life motor systems, making it essential for both academic understanding and practical implementations.