Electrical Circuits

To determine the value of in the given question, we need to compare the electrostatic force to the gravitational force between the ions. The formula for each of these forces is as follows:

1. Electrostatic force between two charges:
   • Where , the Coulomb's constant.
   • and are the charges of ions A and B respectively.
2. Gravitational force between two masses:
   • Where , the gravitational constant.
   • and are the masses of ions A and B respectively.

The ratio of the magnitudes of the electrostatic force to the gravitational force is given by:

Substituting the given values:

Calculating the above expression:

Given , solving for :

Thus, the value of is 10, making the correct answer 10.

The input power to the motor is given by: where is the voltage and is the current. Given and , The efficiency of the motor is given by: Given , we can find the output power: The power loss in the motor is the difference between the input power and the output power: We need to convert the power loss from watts to calories per second (cal/s).
We know that 1 calorie (cal) is equal to 4.184 Joules (J).
Since power is the rate of energy transfer (1 W = 1 J/s), we have: So, the power loss in cal/s is: Rounding to the nearest whole number, the loss of power is approximately 2 cal/s.

To solve this problem, we need to understand the behavior of the circuit which includes a source of voltage, a resistor, and an inductor. The circuit given is an RL circuit, and the rate of change of current is given as 8 A/s when the resistance is 12 Ω. We want to find the current at this instant.

The formula for the induced electromotive force (emf) in an inductor is given by the equation:

where:

• is the induced EMF.
• is the inductance (3 H in this case).
• is the rate of change of current (8 A/s).

Plugging the values into the formula, we get:

In a steady state, the voltage across the resistor and the inductor equals the supply voltage:

Given that the supply voltage is 12 V, and substituting for and , we use the relation:

Rearranging the equation gives:

Thus, the current can be found as:

Therefore, the current in the circuit at the instant described is 3 A.

Step 1: Condition for perpendicularity

Since the vectors are perpendicular, Hence,

Step 2: Condition for equal magnitudes

Given that the magnitudes are equal: Using the magnitude formula:

Step 3: Simplify the equation

Squaring both sides:

Step 4: Substitute into the dot product equation

Substitute into (1): This is a contradiction, so is not valid.

Step 5: Substitute the negative value

Now take :

Final Answer:

Concept: Diode logic circuits use the conducting (forward-biased) and non-conducting (reverse-biased) states of diodes to implement basic logic gates. 
Step 1: Understand the circuit 
A pull-up resistor connects the output node to . 
Diodes and connect inputs and respectively to the output node. 
The diodes are oriented such that a LOW input can pull the output LOW. 

Step 2: Truth table analysis 

Explanation: 
If either or , the corresponding diode conducts and pulls LOW. 
Only when both and , both diodes are reverse-biased and the pull-up resistor makes HIGH. 
Final Answer: