What is the percentage increase in a current required in a coil so that energy stored in the coil becomes doubled?

Explanation:

To determine the percentage increase in the current required in a coil so that the energy stored in the coil becomes doubled, we need to understand the relationship between the energy stored in a coil and the current flowing through it.

The energy (E) stored in an inductor (or a coil) is given by the formula:

E = (1/2) × L × I²

Where:

• E is the energy stored in the inductor (in joules).
• L is the inductance of the coil (in henries).
• I is the current flowing through the coil (in amperes).

Let's assume the initial energy stored in the coil is E₁ and the initial current is I₁. Therefore, we can write:

E₁ = (1/2) × L × I₁²

We need to find the new current I₂ required to double the energy stored in the coil. So, the new energy E₂ will be:

E₂ = 2 × E₁

Substituting the formula for energy in terms of inductance and current, we get:

(1/2) × L × I₂² = 2 × (1/2) × L × I₁²

Simplifying this equation, we get:

L × I₂² = 2 × L × I₁²

Dividing both sides by L, we get:

I₂² = 2 × I₁²

Taking the square root of both sides, we get:

I₂ = √2 × I₁

The percentage increase in the current can be calculated as:

Percentage Increase = [(I₂ - I₁) / I₁] × 100%

Substituting the value of I₂, we get:

Percentage Increase = [(√2 × I₁ - I₁) / I₁] × 100%

Simplifying this, we get:

Percentage Increase = (√2 - 1) × 100%

We know that √2 ≈ 1.414, so:

Percentage Increase = (1.414 - 1) × 100%

Percentage Increase = 0.414 × 100%

Percentage Increase = 41.4%

Therefore, the percentage increase in the current required in the coil so that the energy stored in the coil becomes doubled is 41.4%. Hence, the correct option is Option 3.

Analysis of Other Options:

To further understand why the other options are incorrect, let’s evaluate them:

Option 1: 1%

This option is incorrect because the calculation above shows that the percentage increase in the current is significantly higher than 1%. The actual percentage increase required is 41.4%, which is much larger.

Option 2: 0.5%

This option is also incorrect for similar reasons as Option 1. The percentage increase in the current required is 41.4%, not 0.5%. This value is far too low to result in a doubling of the energy stored in the coil.

Option 3: 0.414

This option is close to the correct answer but seems to be a misinterpretation. While 0.414 is part of the calculation (as √2 - 1 ≈ 0.414), the correct answer should be expressed as a percentage, which is 41.4%.

Option 4: 0.25

This option is incorrect because it does not reflect the correct calculation of the percentage increase in the current needed to double the energy stored in the coil. The actual increase should be 41.4%, not 0.25.

Conclusion:

Understanding the relationship between the energy stored in an inductor and the current flowing through it is essential for solving problems like this. The energy is proportional to the square of the current, and doubling the energy requires the current to be increased by a factor of √2. This results in a 41.4% increase in the current, making Option 3 the correct answer.